HIGH-DIMENSIONAL ELLIPSOIDS CONVERGE TO GAUSSIAN … · CONVERGE TO GAUSSIAN SPACES DAISUKE...
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HIGH-DIMENSIONAL ELLIPSOIDS
CONVERGE TO GAUSSIAN SPACES
DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Abstract We prove the convergence of (solid) ellipsoids to aGaussian space in Gromovrsquos concentrationweak topology as thedimension diverges to infinity This gives the first discovered ex-ample of an irreducible nontrivial convergent sequence in the con-centration topology where lsquoirreducible nontrivialrsquo roughly meansto be not constructed from Levy families nor box convergent se-quences
1 Introduction
The study of convergence of metric measure spaces is one of cen-tral topics in geometric analysis on metric measure spaces We referto [8 9 14 23 24] for some celebrated works on it Such the studyoriginates that of Gromov-Hausdorff convergencecollapsing of Rie-mannian manifolds which has widely been developed and applied tosolutions to many significant problems in geometry and topology in-cluding Thurstonrsquos geometrization conjecture [21 28] As the startingpoint of geometric analytic study in the collapsing theory Fukaya [5]introduced the concept of measured Gromov-Hausdorff convergence ofmetric measure spaces to study the Laplacian of collapsing Riemann-ian manifolds There he discovered that only the metric structure butalso the measure structure plays an important role in the collapsingphenomena After that Cheeger-Colding [2ndash4] established a theoryof measured Gromov-Hausdorff limits of complete Riemannian mani-folds with a lower bound of Ricci curvature which is nowadays widelyapplied in the Riemannian and Kahler geometry
Meanwhile Gromov [9 Chapter 312+
] (see also [25]) has developed anew convergence theory of metric measure spaces based on the concen-tration of measure phenomenon due to Levy and V Milman [121315]where the concentration of measure phenomenon is roughly stated asthat any 1-Lipschitz function on high-dimensional spaces is almost con-stant In Gromovrsquos theory he introduced two fundamental concepts of
Date March 12 20202010 Mathematics Subject Classification 53C23Key words and phrases ellipsoid concentration topology Gaussian space ob-
servable distance box distance pyramidThis work was supported by JSPS KAKENHI Grant Number 19K03459 and
17J021211
2 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
distance functions the observable distance function dconc and the box
distance function on the set say X of isomorphism classes of metricmeasure spaces The box distance function is nearly a metrization ofmeasured Gromov-Hausdorff convergence (precisely the isomorphismclasses are little different) while the observable distance function in-duces a very characteristic topology called the concentration topologywhich is effective to capture the high-dimensional aspects of spacesThe concentration topology is weaker than the box topology and inparticular a measured Gromov-Hausdorff convergence becomes a con-vergence in the concentration topology He also introduced a naturalcompactification say Π of X with respect to the concentration topol-ogy where the topology on Π is called the weak topology The concen-tration topology is sometimes useful to investigate the dimension-freeproperties of manifolds For example it has been applied to obtain anew dimension-free estimate of eigenvalue ratios of the drifted Lapla-cian on a closed Riemannian manifold with nonnegative Bakry-EmeryRicci curvature [6]
The study of the concentration and weak topologies has been growingrapidly in recent years (see [6101116ndash202225ndash27]) However thereare only a few nontrivial examples of convergent sequences of met-ric measure spaces in the concentration and weak topologies wherelsquonontrivialrsquo means neither to be a Levy family (ie convergent to aone-point space) to infinitely dissipate (see Subsection 26 for dissi-pation) nor to be box convergent One way to construct a nontriv-ial convergent sequence is to take the disjoint union or the product(more generally the fibration) of trivial sequences and to perform lit-tle surgery on it (and also to repeat these procedures finitely manytimes) We call a sequence obtained in this way a reducible sequenceAn irreducible sequence is a sequence that is not reducible In thispaper any sequence of (solid) ellipsoids has a subsequence convergingto an infinite-dimensional Gaussian space in the concentrationweaktopology This provides a new family of nontrivial weak convergentsequences and especially contains the first discovered example of anirreducible nontrivial sequence that is convergent in the concentrationtopology
Let us state our main results precisely A solid ellipsoid and anellipsoid are respectively written as
Enαi = x isin Rn |nsum
i=1
x2iα2i
le 1
Snminus1αi = x isin Rn |
nsum
i=1
x2iα2i
= 1
HIGH-DIMENSIONAL ELLIPSOIDS 3
where αi i = 1 2 n is a finite sequence of positive real num-bers See Section 3 for the definition of their metric-measure struc-tures Denote by En
αi any one of Enαi and Snminus1αi Let us given a
sequence En(j)αijij of (solid) ellipsoids where αij i = 1 2 n(j)
j = 1 2 is a double sequence of positive real numbers Our prob-lem is to determine under what condition it will converge in the con-centrationweak topology and to describe its limit
In the case where the dimension n(j) is bounded for all j the problemis easy to solve In fact such the sequence has a Hausdorff-convergentsubsequence in a Euclidean space which is also box convergent if αijis bounded for all i and j the sequence has an infinitely dissipatingsubsequence if αij is unbounded
We set aij = αijradicn(j)minus 1 If n(j) and supi aij both diverge to
infinity as j rarr infin then it is also easy to prove that Enαi infinitely
dissipates (see Proposition 33)By the reason we have mentioned above we assume
(A0) n(j) diverges to infinity as j rarr infin and aij is bounded for all iand j
We further consider the following three conditions
(A1) n(j) is monotone nondecreasing in j(A2) aij is monotone nonincreasing in i for each j(A3) aij converges to a real number say ai as j rarr infin for each i
Note that (A2) and (A3) together imply that ai is monotone nonin-creasing in i
Any sequence of (solid) ellipsoids with (A0) contains a subsequence
Ej such that each Ej is isomorphic to En(j)
radicn(j)minus1 aiji
for some se-
quence aij satisfying (A0)ndash(A3) In fact we have a subsequencefor which the dimensions satisfy (A1) Then exchanging the axes ofcoordinate provides (A2) A diagonal argument proves to have a sub-sequence satisfying (A3) Thus our problem becomes to investigate
the convergence of En(j)
radicn(j)minus1 aiji
j satisfying (A0)ndash(A3)
One of our main theorems is stated as follows Refer to Subsection28 for the definition of the Gaussian space Γinfin
a2i
Theorem 11 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers satisfying (A0)ndash(A3) Then En(j)
radicn(j)minus1 aiji
converges weakly to the infinite-dimensional Gaussian space Γinfina2
i as
j rarr infin This convergence becomes a convergence in the concentration
topology if and only if ai is an l2-sequence Moreover this conver-
gence becomes an asymptotic concentration (ie a dconc-Cauchy se-
quence) if and only if ai converges to zero
4 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Gromov presents an exercise [9 31257] which is some easier special
cases of our theorem Our theorem provides not only an answer butalso a complete generalization of his exercise
For the case of round spheres (ie aij = a1j for all i and j) and alsoof projective spaces the theorem is formerly obtained by the secondnamed author [25 26] for which the convergence is only weak Alsothe weak convergence of Stiefel and flag manifolds are studied jointlyby Takatsu and the second named author [27]
We emphasize that convergence in the weakconcentration topol-ogy is completely different from weak convergence of measures Forinstance the Prokhorov distance between the normalized volume mea-sure on Snminus1(
radicnminus 1) and the n-dimensional standard Gaussian mea-
sure on Rn is bounded away from zero [27] though they both convergeto the infinite-dimensional standard Gaussian space in Gromovrsquos weaktopology
As for the characterization of weak convergence of measures weprove in Proposition 42 that if aiji l2-converges to an l2-sequence
ai as j rarr infin then the measure of En(j)
radicn(j)minus1 aiji
converges weakly to
the Gaussian measure γinfinai on a Hilbert space and consequently theweak convergence in Theorem 11 becomes the box convergence Con-versely the l2-convergence of aiji is also a necessary condition forthe box convergence of the (solid) ellipsoids as is seen in the followingtheorem
Theorem 12 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers satisfying (A0)ndash(A3) Then the convergence
in Theorem 11 becomes a box convergence if and only if we have
infinsum
i=1
a2i lt +infin and limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Theorems 11 and 12 together provide an example of irreduciblenontrivial convergent sequence of metric measure spaces in the con-centration topology ie the sequence of the (solid) ellipsoids with anl2-sequence ai and with a non-l2-convergent aiji as j rarr infin
The proof of the lsquoonly ifrsquo part of Theorem 12 is highly nontrivial Ifaiji does not l2-converge then it is easy to see that the measure ofthe (solid) ellipsoid in such the sequence does not converge weakly inthe Hilbert space However this is not enough to obtain the box non-convergence because we consider the isomorphism classes of (solid)ellipsoids for the box convergence For the complete proof we need adelicate discussion using Theorem 11
Let us briefly mention the outline of the proof of the weak con-vergence of solid ellipsoids in Theorem 11 For simplicity we set
HIGH-DIMENSIONAL ELLIPSOIDS 5
En = En(j)
radicn(j)minus1 aiji
and Γ = Γinfina2i
For the weak convergence it
is sufficient to show that
(11) the limit of En dominates Γ(12) Γ dominates the limit of En
where for two metric measure spaces X and Y the space X dominates
Y if there is a 1-Lipschitz map from X to Y preserving their measures(11) easily follows from the Maxwell-Boltzmann distribution law
(Proposition 32)(12) is much harder to prove Let us first consider the simple case
where En is the ball Bn(radicnminus 1) of radius
radicnminus 1 and where Γ = Γinfin
12
We see that for any fixed 0 lt θ lt 1 the n-dimensional Gaussian mea-sure γn12 and the normalized volume measure of Bn(θ
radicnminus 1) both are
very small for large n Ignoring this small part Bn(θradicnminus 1) we find
a measure-preserving isotropic map say ϕ from Γn12 Bn(θradicnminus 1)
to the annulus Bn(radicnminus 1) Bn(θ
radicnminus 1) where we normalize their
measures to be probability Estimating the Lipschitz constant of ϕwe obtain (12) with error This error is estimated and we eventuallyobtain the required weak convergence
We next try to apply this discussion for solid ellipsoids We considerthe distortion of the above isotropic map ϕ by a linear transforma-tion determined by aij However the Lipschitz constant of such thedistorted isotropic map is arbitrarily large depending on aij To over-come this problem we settle the assumptions (A0)ndash(A3) from whichthe discussion boils down to the special case where ai = aN for alli ge N and aij = ai ge aN for all i j and for a (large) number N Infact by (A0)ndash(A3) the solid ellipsoid En for large n and the Gaussianspace Γ are both close to those in the above special case In this specialcase the Gaussian measure γna2i
and the normalized volume measure
of En of the domain
x isin Rn o | |xi|x lt ε for any i = 1 N minus 1
are both almost full for large n and for any fixed ε gt 0 On this domainwe are able to estimate the Lipschitz constant of the distorted isotropicmap With some careful error estimates letting ǫ rarr 0+ and θ rarr 1minuswe prove the weak convergence of En to Γ
2 Preliminaries
In this section we survey the definitions and the facts needed in thispaper We refer to [9 Chapter 31
2+] and [25] for more details
21 Distance between measures
6 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Definition 21 (Total variation distance) The total variation distance
dTV(micro ν) of two Borel probability measures micro and ν on a topologicalspace X is defined by
dTV(micro ν) = supA
|micro(A)minus ν(A) |
where A runs over all Borel subsets of X
If micro and ν are both absolutely continuous with respect to a Borelmeasure ω on X then
dTV(micro ν) =1
2
int
X
∣∣∣∣dmicro
dωminus dν
dω
∣∣∣∣ dω
where dmicrodω
is the Radon-Nikodym derivative of micro with respect to ω
Definition 22 (Prokhorov distance) The Prokhorov distance dP(micro ν)between two Borel probability measures micro and ν on a metric space(X dX) is defined to be the infimum of ε ge 0 satisfying
micro(Bε(A)) ge ν(A)minus ε
for any Borel subset A sub X where Bε(A) = x isin X | dX(xA) lt ε The Prokhorov metric is a metrization of weak convergence of Borel
probability measures on X provided thatX is a separable metric spaceIt is known that dP le dTV
Definition 23 (Ky Fan distance) Let (X micro) be a measure space andY a metric space For two micro-measurable maps f g X rarr Y we definethe Ky Fan distance dKF(f g) between f and g to be the infimum ofε ge 0 satisfying
micro( x isin X | dY (f(x) g(x)) gt ε ) le ε
dKF is a pseudo-metric on the set of micro-measurable maps from X toY It holds that dKF(f g) = 0 if and only if f = g micro-ae We havedP(flowastmicro glowastmicro) le dKF(f g) where flowastmicro is the push-forward of micro by f
Let p be a real number with p ge 1 and (X dX) a complete separablemetric space
Definition 24 The p-Wasserstein distance between two Borel prob-ability measures micro and ν on X is defined to be
Wp(micro ν) = infπisinΠ(microν)
(int
XtimesXdX(x x
prime)p dπ(x xprime)
) 1p
(le +infin)
where Π(micro ν) is the set of couplings between micro and ν ie the set ofBorel probability measures π on X times X such that π(A times X) = micro(A)and π(X times A) = ν(A) for any Borel subset A sub X
Lemma 25 Let micro and micron n = 1 2 be Borel probability measures
on X Then the following (1) and (2) are equivalent to each other
(1) Wp(micron micro) rarr 0 as nrarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 7
(2) micron converges weakly to micro as nrarr infin and the p-th moment of micronis uniformly bounded
lim supnrarrinfin
int
X
dX(x0 x)p dmicron(x) lt +infin
for some point x0 isin X
It is known that dP2 leW1 and that Wp leWq for any 1 le p le q
22 mm-Isomorphism and Lipschitz order
Definition 26 (mm-Space) Let (X dX) be a complete separable met-ric space and microX a Borel probability measure on X We call the triple(X dX microX) an mm-space We sometimes say that X is an mm-spacein which case the metric and the Borel measure of X are respectivelyindicated by dX and microX
Definition 27 (mm-Isomorphism) Two mm-spaces X and Y are saidto be mm-isomorphic to each other if there exists an isometry f supp microX rarr supp microY with flowastmicroX = microY where supp microX is the support ofmicroX Such an isometry f is called an mm-isomorphism Denote by Xthe set of mm-isomorphism classes of mm-spaces
Note that X is mm-isomorphic to (supp microX dX microX)We assume that an mm-space X satisfies
X = supp microX
unless otherwise stated
Definition 28 (Lipschitz order) Let X and Y be two mm-spaces Wesay that X (Lipschitz ) dominates Y and write Y ≺ X if there exists a1-Lipschitz map f X rarr Y satisfying flowastmicroX = microY We call the relation≺ on X the Lipschitz order
The Lipschitz order ≺ is a partial order relation on X
23 Observable diameter The observable diameter is one of themost fundamental invariants of an mm-space up to mm-isomorphism
Definition 29 (Partial and observable diameter) Let X be an mm-space and let κ gt 0 We define the κ-partial diameter diam(X 1minusκ) =diam(microX 1 minus κ) of X to be the infimum of the diameter of A whereA sub X runs over all Borel subsets with microX(A) ge 1 minus κ Denote byLip1(X) the set of 1-Lipschitz continuous real-valued functions on X We define the (κ-)observable diameter of X by
ObsDiam(X minusκ) = supfisinLip1(X)
diam(flowastmicroX 1minus κ)
ObsDiam(X) = infκgt0
maxObsDiam(X minusκ) κ
It is easy to see that the (κ-)observable diameter is monotone non-decreasing with respect to the Lipschitz order relation
8 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
24 Box distance and observable distance
Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I
It is known that any mm-space has a parameter
Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that
L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε
for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I
The box metric is a complete separable metric on X
Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that
(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x
prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε
We call the set X a nonexceptional domain of f
Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le
3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from
X to Y
Definition 214 (Observable distance) For any parameter ϕ of X weset
ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces
X and Y by
dconc(X Y ) = infϕψ
dH(ϕlowastLip1(X) ψlowastLip1(Y ))
where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X
It is known that dconc le and that the concentration topology isweaker than the box topology
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
2 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
distance functions the observable distance function dconc and the box
distance function on the set say X of isomorphism classes of metricmeasure spaces The box distance function is nearly a metrization ofmeasured Gromov-Hausdorff convergence (precisely the isomorphismclasses are little different) while the observable distance function in-duces a very characteristic topology called the concentration topologywhich is effective to capture the high-dimensional aspects of spacesThe concentration topology is weaker than the box topology and inparticular a measured Gromov-Hausdorff convergence becomes a con-vergence in the concentration topology He also introduced a naturalcompactification say Π of X with respect to the concentration topol-ogy where the topology on Π is called the weak topology The concen-tration topology is sometimes useful to investigate the dimension-freeproperties of manifolds For example it has been applied to obtain anew dimension-free estimate of eigenvalue ratios of the drifted Lapla-cian on a closed Riemannian manifold with nonnegative Bakry-EmeryRicci curvature [6]
The study of the concentration and weak topologies has been growingrapidly in recent years (see [6101116ndash202225ndash27]) However thereare only a few nontrivial examples of convergent sequences of met-ric measure spaces in the concentration and weak topologies wherelsquonontrivialrsquo means neither to be a Levy family (ie convergent to aone-point space) to infinitely dissipate (see Subsection 26 for dissi-pation) nor to be box convergent One way to construct a nontriv-ial convergent sequence is to take the disjoint union or the product(more generally the fibration) of trivial sequences and to perform lit-tle surgery on it (and also to repeat these procedures finitely manytimes) We call a sequence obtained in this way a reducible sequenceAn irreducible sequence is a sequence that is not reducible In thispaper any sequence of (solid) ellipsoids has a subsequence convergingto an infinite-dimensional Gaussian space in the concentrationweaktopology This provides a new family of nontrivial weak convergentsequences and especially contains the first discovered example of anirreducible nontrivial sequence that is convergent in the concentrationtopology
Let us state our main results precisely A solid ellipsoid and anellipsoid are respectively written as
Enαi = x isin Rn |nsum
i=1
x2iα2i
le 1
Snminus1αi = x isin Rn |
nsum
i=1
x2iα2i
= 1
HIGH-DIMENSIONAL ELLIPSOIDS 3
where αi i = 1 2 n is a finite sequence of positive real num-bers See Section 3 for the definition of their metric-measure struc-tures Denote by En
αi any one of Enαi and Snminus1αi Let us given a
sequence En(j)αijij of (solid) ellipsoids where αij i = 1 2 n(j)
j = 1 2 is a double sequence of positive real numbers Our prob-lem is to determine under what condition it will converge in the con-centrationweak topology and to describe its limit
In the case where the dimension n(j) is bounded for all j the problemis easy to solve In fact such the sequence has a Hausdorff-convergentsubsequence in a Euclidean space which is also box convergent if αijis bounded for all i and j the sequence has an infinitely dissipatingsubsequence if αij is unbounded
We set aij = αijradicn(j)minus 1 If n(j) and supi aij both diverge to
infinity as j rarr infin then it is also easy to prove that Enαi infinitely
dissipates (see Proposition 33)By the reason we have mentioned above we assume
(A0) n(j) diverges to infinity as j rarr infin and aij is bounded for all iand j
We further consider the following three conditions
(A1) n(j) is monotone nondecreasing in j(A2) aij is monotone nonincreasing in i for each j(A3) aij converges to a real number say ai as j rarr infin for each i
Note that (A2) and (A3) together imply that ai is monotone nonin-creasing in i
Any sequence of (solid) ellipsoids with (A0) contains a subsequence
Ej such that each Ej is isomorphic to En(j)
radicn(j)minus1 aiji
for some se-
quence aij satisfying (A0)ndash(A3) In fact we have a subsequencefor which the dimensions satisfy (A1) Then exchanging the axes ofcoordinate provides (A2) A diagonal argument proves to have a sub-sequence satisfying (A3) Thus our problem becomes to investigate
the convergence of En(j)
radicn(j)minus1 aiji
j satisfying (A0)ndash(A3)
One of our main theorems is stated as follows Refer to Subsection28 for the definition of the Gaussian space Γinfin
a2i
Theorem 11 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers satisfying (A0)ndash(A3) Then En(j)
radicn(j)minus1 aiji
converges weakly to the infinite-dimensional Gaussian space Γinfina2
i as
j rarr infin This convergence becomes a convergence in the concentration
topology if and only if ai is an l2-sequence Moreover this conver-
gence becomes an asymptotic concentration (ie a dconc-Cauchy se-
quence) if and only if ai converges to zero
4 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Gromov presents an exercise [9 31257] which is some easier special
cases of our theorem Our theorem provides not only an answer butalso a complete generalization of his exercise
For the case of round spheres (ie aij = a1j for all i and j) and alsoof projective spaces the theorem is formerly obtained by the secondnamed author [25 26] for which the convergence is only weak Alsothe weak convergence of Stiefel and flag manifolds are studied jointlyby Takatsu and the second named author [27]
We emphasize that convergence in the weakconcentration topol-ogy is completely different from weak convergence of measures Forinstance the Prokhorov distance between the normalized volume mea-sure on Snminus1(
radicnminus 1) and the n-dimensional standard Gaussian mea-
sure on Rn is bounded away from zero [27] though they both convergeto the infinite-dimensional standard Gaussian space in Gromovrsquos weaktopology
As for the characterization of weak convergence of measures weprove in Proposition 42 that if aiji l2-converges to an l2-sequence
ai as j rarr infin then the measure of En(j)
radicn(j)minus1 aiji
converges weakly to
the Gaussian measure γinfinai on a Hilbert space and consequently theweak convergence in Theorem 11 becomes the box convergence Con-versely the l2-convergence of aiji is also a necessary condition forthe box convergence of the (solid) ellipsoids as is seen in the followingtheorem
Theorem 12 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers satisfying (A0)ndash(A3) Then the convergence
in Theorem 11 becomes a box convergence if and only if we have
infinsum
i=1
a2i lt +infin and limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Theorems 11 and 12 together provide an example of irreduciblenontrivial convergent sequence of metric measure spaces in the con-centration topology ie the sequence of the (solid) ellipsoids with anl2-sequence ai and with a non-l2-convergent aiji as j rarr infin
The proof of the lsquoonly ifrsquo part of Theorem 12 is highly nontrivial Ifaiji does not l2-converge then it is easy to see that the measure ofthe (solid) ellipsoid in such the sequence does not converge weakly inthe Hilbert space However this is not enough to obtain the box non-convergence because we consider the isomorphism classes of (solid)ellipsoids for the box convergence For the complete proof we need adelicate discussion using Theorem 11
Let us briefly mention the outline of the proof of the weak con-vergence of solid ellipsoids in Theorem 11 For simplicity we set
HIGH-DIMENSIONAL ELLIPSOIDS 5
En = En(j)
radicn(j)minus1 aiji
and Γ = Γinfina2i
For the weak convergence it
is sufficient to show that
(11) the limit of En dominates Γ(12) Γ dominates the limit of En
where for two metric measure spaces X and Y the space X dominates
Y if there is a 1-Lipschitz map from X to Y preserving their measures(11) easily follows from the Maxwell-Boltzmann distribution law
(Proposition 32)(12) is much harder to prove Let us first consider the simple case
where En is the ball Bn(radicnminus 1) of radius
radicnminus 1 and where Γ = Γinfin
12
We see that for any fixed 0 lt θ lt 1 the n-dimensional Gaussian mea-sure γn12 and the normalized volume measure of Bn(θ
radicnminus 1) both are
very small for large n Ignoring this small part Bn(θradicnminus 1) we find
a measure-preserving isotropic map say ϕ from Γn12 Bn(θradicnminus 1)
to the annulus Bn(radicnminus 1) Bn(θ
radicnminus 1) where we normalize their
measures to be probability Estimating the Lipschitz constant of ϕwe obtain (12) with error This error is estimated and we eventuallyobtain the required weak convergence
We next try to apply this discussion for solid ellipsoids We considerthe distortion of the above isotropic map ϕ by a linear transforma-tion determined by aij However the Lipschitz constant of such thedistorted isotropic map is arbitrarily large depending on aij To over-come this problem we settle the assumptions (A0)ndash(A3) from whichthe discussion boils down to the special case where ai = aN for alli ge N and aij = ai ge aN for all i j and for a (large) number N Infact by (A0)ndash(A3) the solid ellipsoid En for large n and the Gaussianspace Γ are both close to those in the above special case In this specialcase the Gaussian measure γna2i
and the normalized volume measure
of En of the domain
x isin Rn o | |xi|x lt ε for any i = 1 N minus 1
are both almost full for large n and for any fixed ε gt 0 On this domainwe are able to estimate the Lipschitz constant of the distorted isotropicmap With some careful error estimates letting ǫ rarr 0+ and θ rarr 1minuswe prove the weak convergence of En to Γ
2 Preliminaries
In this section we survey the definitions and the facts needed in thispaper We refer to [9 Chapter 31
2+] and [25] for more details
21 Distance between measures
6 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Definition 21 (Total variation distance) The total variation distance
dTV(micro ν) of two Borel probability measures micro and ν on a topologicalspace X is defined by
dTV(micro ν) = supA
|micro(A)minus ν(A) |
where A runs over all Borel subsets of X
If micro and ν are both absolutely continuous with respect to a Borelmeasure ω on X then
dTV(micro ν) =1
2
int
X
∣∣∣∣dmicro
dωminus dν
dω
∣∣∣∣ dω
where dmicrodω
is the Radon-Nikodym derivative of micro with respect to ω
Definition 22 (Prokhorov distance) The Prokhorov distance dP(micro ν)between two Borel probability measures micro and ν on a metric space(X dX) is defined to be the infimum of ε ge 0 satisfying
micro(Bε(A)) ge ν(A)minus ε
for any Borel subset A sub X where Bε(A) = x isin X | dX(xA) lt ε The Prokhorov metric is a metrization of weak convergence of Borel
probability measures on X provided thatX is a separable metric spaceIt is known that dP le dTV
Definition 23 (Ky Fan distance) Let (X micro) be a measure space andY a metric space For two micro-measurable maps f g X rarr Y we definethe Ky Fan distance dKF(f g) between f and g to be the infimum ofε ge 0 satisfying
micro( x isin X | dY (f(x) g(x)) gt ε ) le ε
dKF is a pseudo-metric on the set of micro-measurable maps from X toY It holds that dKF(f g) = 0 if and only if f = g micro-ae We havedP(flowastmicro glowastmicro) le dKF(f g) where flowastmicro is the push-forward of micro by f
Let p be a real number with p ge 1 and (X dX) a complete separablemetric space
Definition 24 The p-Wasserstein distance between two Borel prob-ability measures micro and ν on X is defined to be
Wp(micro ν) = infπisinΠ(microν)
(int
XtimesXdX(x x
prime)p dπ(x xprime)
) 1p
(le +infin)
where Π(micro ν) is the set of couplings between micro and ν ie the set ofBorel probability measures π on X times X such that π(A times X) = micro(A)and π(X times A) = ν(A) for any Borel subset A sub X
Lemma 25 Let micro and micron n = 1 2 be Borel probability measures
on X Then the following (1) and (2) are equivalent to each other
(1) Wp(micron micro) rarr 0 as nrarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 7
(2) micron converges weakly to micro as nrarr infin and the p-th moment of micronis uniformly bounded
lim supnrarrinfin
int
X
dX(x0 x)p dmicron(x) lt +infin
for some point x0 isin X
It is known that dP2 leW1 and that Wp leWq for any 1 le p le q
22 mm-Isomorphism and Lipschitz order
Definition 26 (mm-Space) Let (X dX) be a complete separable met-ric space and microX a Borel probability measure on X We call the triple(X dX microX) an mm-space We sometimes say that X is an mm-spacein which case the metric and the Borel measure of X are respectivelyindicated by dX and microX
Definition 27 (mm-Isomorphism) Two mm-spaces X and Y are saidto be mm-isomorphic to each other if there exists an isometry f supp microX rarr supp microY with flowastmicroX = microY where supp microX is the support ofmicroX Such an isometry f is called an mm-isomorphism Denote by Xthe set of mm-isomorphism classes of mm-spaces
Note that X is mm-isomorphic to (supp microX dX microX)We assume that an mm-space X satisfies
X = supp microX
unless otherwise stated
Definition 28 (Lipschitz order) Let X and Y be two mm-spaces Wesay that X (Lipschitz ) dominates Y and write Y ≺ X if there exists a1-Lipschitz map f X rarr Y satisfying flowastmicroX = microY We call the relation≺ on X the Lipschitz order
The Lipschitz order ≺ is a partial order relation on X
23 Observable diameter The observable diameter is one of themost fundamental invariants of an mm-space up to mm-isomorphism
Definition 29 (Partial and observable diameter) Let X be an mm-space and let κ gt 0 We define the κ-partial diameter diam(X 1minusκ) =diam(microX 1 minus κ) of X to be the infimum of the diameter of A whereA sub X runs over all Borel subsets with microX(A) ge 1 minus κ Denote byLip1(X) the set of 1-Lipschitz continuous real-valued functions on X We define the (κ-)observable diameter of X by
ObsDiam(X minusκ) = supfisinLip1(X)
diam(flowastmicroX 1minus κ)
ObsDiam(X) = infκgt0
maxObsDiam(X minusκ) κ
It is easy to see that the (κ-)observable diameter is monotone non-decreasing with respect to the Lipschitz order relation
8 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
24 Box distance and observable distance
Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I
It is known that any mm-space has a parameter
Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that
L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε
for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I
The box metric is a complete separable metric on X
Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that
(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x
prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε
We call the set X a nonexceptional domain of f
Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le
3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from
X to Y
Definition 214 (Observable distance) For any parameter ϕ of X weset
ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces
X and Y by
dconc(X Y ) = infϕψ
dH(ϕlowastLip1(X) ψlowastLip1(Y ))
where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X
It is known that dconc le and that the concentration topology isweaker than the box topology
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
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infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 3
where αi i = 1 2 n is a finite sequence of positive real num-bers See Section 3 for the definition of their metric-measure struc-tures Denote by En
αi any one of Enαi and Snminus1αi Let us given a
sequence En(j)αijij of (solid) ellipsoids where αij i = 1 2 n(j)
j = 1 2 is a double sequence of positive real numbers Our prob-lem is to determine under what condition it will converge in the con-centrationweak topology and to describe its limit
In the case where the dimension n(j) is bounded for all j the problemis easy to solve In fact such the sequence has a Hausdorff-convergentsubsequence in a Euclidean space which is also box convergent if αijis bounded for all i and j the sequence has an infinitely dissipatingsubsequence if αij is unbounded
We set aij = αijradicn(j)minus 1 If n(j) and supi aij both diverge to
infinity as j rarr infin then it is also easy to prove that Enαi infinitely
dissipates (see Proposition 33)By the reason we have mentioned above we assume
(A0) n(j) diverges to infinity as j rarr infin and aij is bounded for all iand j
We further consider the following three conditions
(A1) n(j) is monotone nondecreasing in j(A2) aij is monotone nonincreasing in i for each j(A3) aij converges to a real number say ai as j rarr infin for each i
Note that (A2) and (A3) together imply that ai is monotone nonin-creasing in i
Any sequence of (solid) ellipsoids with (A0) contains a subsequence
Ej such that each Ej is isomorphic to En(j)
radicn(j)minus1 aiji
for some se-
quence aij satisfying (A0)ndash(A3) In fact we have a subsequencefor which the dimensions satisfy (A1) Then exchanging the axes ofcoordinate provides (A2) A diagonal argument proves to have a sub-sequence satisfying (A3) Thus our problem becomes to investigate
the convergence of En(j)
radicn(j)minus1 aiji
j satisfying (A0)ndash(A3)
One of our main theorems is stated as follows Refer to Subsection28 for the definition of the Gaussian space Γinfin
a2i
Theorem 11 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers satisfying (A0)ndash(A3) Then En(j)
radicn(j)minus1 aiji
converges weakly to the infinite-dimensional Gaussian space Γinfina2
i as
j rarr infin This convergence becomes a convergence in the concentration
topology if and only if ai is an l2-sequence Moreover this conver-
gence becomes an asymptotic concentration (ie a dconc-Cauchy se-
quence) if and only if ai converges to zero
4 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Gromov presents an exercise [9 31257] which is some easier special
cases of our theorem Our theorem provides not only an answer butalso a complete generalization of his exercise
For the case of round spheres (ie aij = a1j for all i and j) and alsoof projective spaces the theorem is formerly obtained by the secondnamed author [25 26] for which the convergence is only weak Alsothe weak convergence of Stiefel and flag manifolds are studied jointlyby Takatsu and the second named author [27]
We emphasize that convergence in the weakconcentration topol-ogy is completely different from weak convergence of measures Forinstance the Prokhorov distance between the normalized volume mea-sure on Snminus1(
radicnminus 1) and the n-dimensional standard Gaussian mea-
sure on Rn is bounded away from zero [27] though they both convergeto the infinite-dimensional standard Gaussian space in Gromovrsquos weaktopology
As for the characterization of weak convergence of measures weprove in Proposition 42 that if aiji l2-converges to an l2-sequence
ai as j rarr infin then the measure of En(j)
radicn(j)minus1 aiji
converges weakly to
the Gaussian measure γinfinai on a Hilbert space and consequently theweak convergence in Theorem 11 becomes the box convergence Con-versely the l2-convergence of aiji is also a necessary condition forthe box convergence of the (solid) ellipsoids as is seen in the followingtheorem
Theorem 12 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers satisfying (A0)ndash(A3) Then the convergence
in Theorem 11 becomes a box convergence if and only if we have
infinsum
i=1
a2i lt +infin and limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Theorems 11 and 12 together provide an example of irreduciblenontrivial convergent sequence of metric measure spaces in the con-centration topology ie the sequence of the (solid) ellipsoids with anl2-sequence ai and with a non-l2-convergent aiji as j rarr infin
The proof of the lsquoonly ifrsquo part of Theorem 12 is highly nontrivial Ifaiji does not l2-converge then it is easy to see that the measure ofthe (solid) ellipsoid in such the sequence does not converge weakly inthe Hilbert space However this is not enough to obtain the box non-convergence because we consider the isomorphism classes of (solid)ellipsoids for the box convergence For the complete proof we need adelicate discussion using Theorem 11
Let us briefly mention the outline of the proof of the weak con-vergence of solid ellipsoids in Theorem 11 For simplicity we set
HIGH-DIMENSIONAL ELLIPSOIDS 5
En = En(j)
radicn(j)minus1 aiji
and Γ = Γinfina2i
For the weak convergence it
is sufficient to show that
(11) the limit of En dominates Γ(12) Γ dominates the limit of En
where for two metric measure spaces X and Y the space X dominates
Y if there is a 1-Lipschitz map from X to Y preserving their measures(11) easily follows from the Maxwell-Boltzmann distribution law
(Proposition 32)(12) is much harder to prove Let us first consider the simple case
where En is the ball Bn(radicnminus 1) of radius
radicnminus 1 and where Γ = Γinfin
12
We see that for any fixed 0 lt θ lt 1 the n-dimensional Gaussian mea-sure γn12 and the normalized volume measure of Bn(θ
radicnminus 1) both are
very small for large n Ignoring this small part Bn(θradicnminus 1) we find
a measure-preserving isotropic map say ϕ from Γn12 Bn(θradicnminus 1)
to the annulus Bn(radicnminus 1) Bn(θ
radicnminus 1) where we normalize their
measures to be probability Estimating the Lipschitz constant of ϕwe obtain (12) with error This error is estimated and we eventuallyobtain the required weak convergence
We next try to apply this discussion for solid ellipsoids We considerthe distortion of the above isotropic map ϕ by a linear transforma-tion determined by aij However the Lipschitz constant of such thedistorted isotropic map is arbitrarily large depending on aij To over-come this problem we settle the assumptions (A0)ndash(A3) from whichthe discussion boils down to the special case where ai = aN for alli ge N and aij = ai ge aN for all i j and for a (large) number N Infact by (A0)ndash(A3) the solid ellipsoid En for large n and the Gaussianspace Γ are both close to those in the above special case In this specialcase the Gaussian measure γna2i
and the normalized volume measure
of En of the domain
x isin Rn o | |xi|x lt ε for any i = 1 N minus 1
are both almost full for large n and for any fixed ε gt 0 On this domainwe are able to estimate the Lipschitz constant of the distorted isotropicmap With some careful error estimates letting ǫ rarr 0+ and θ rarr 1minuswe prove the weak convergence of En to Γ
2 Preliminaries
In this section we survey the definitions and the facts needed in thispaper We refer to [9 Chapter 31
2+] and [25] for more details
21 Distance between measures
6 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Definition 21 (Total variation distance) The total variation distance
dTV(micro ν) of two Borel probability measures micro and ν on a topologicalspace X is defined by
dTV(micro ν) = supA
|micro(A)minus ν(A) |
where A runs over all Borel subsets of X
If micro and ν are both absolutely continuous with respect to a Borelmeasure ω on X then
dTV(micro ν) =1
2
int
X
∣∣∣∣dmicro
dωminus dν
dω
∣∣∣∣ dω
where dmicrodω
is the Radon-Nikodym derivative of micro with respect to ω
Definition 22 (Prokhorov distance) The Prokhorov distance dP(micro ν)between two Borel probability measures micro and ν on a metric space(X dX) is defined to be the infimum of ε ge 0 satisfying
micro(Bε(A)) ge ν(A)minus ε
for any Borel subset A sub X where Bε(A) = x isin X | dX(xA) lt ε The Prokhorov metric is a metrization of weak convergence of Borel
probability measures on X provided thatX is a separable metric spaceIt is known that dP le dTV
Definition 23 (Ky Fan distance) Let (X micro) be a measure space andY a metric space For two micro-measurable maps f g X rarr Y we definethe Ky Fan distance dKF(f g) between f and g to be the infimum ofε ge 0 satisfying
micro( x isin X | dY (f(x) g(x)) gt ε ) le ε
dKF is a pseudo-metric on the set of micro-measurable maps from X toY It holds that dKF(f g) = 0 if and only if f = g micro-ae We havedP(flowastmicro glowastmicro) le dKF(f g) where flowastmicro is the push-forward of micro by f
Let p be a real number with p ge 1 and (X dX) a complete separablemetric space
Definition 24 The p-Wasserstein distance between two Borel prob-ability measures micro and ν on X is defined to be
Wp(micro ν) = infπisinΠ(microν)
(int
XtimesXdX(x x
prime)p dπ(x xprime)
) 1p
(le +infin)
where Π(micro ν) is the set of couplings between micro and ν ie the set ofBorel probability measures π on X times X such that π(A times X) = micro(A)and π(X times A) = ν(A) for any Borel subset A sub X
Lemma 25 Let micro and micron n = 1 2 be Borel probability measures
on X Then the following (1) and (2) are equivalent to each other
(1) Wp(micron micro) rarr 0 as nrarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 7
(2) micron converges weakly to micro as nrarr infin and the p-th moment of micronis uniformly bounded
lim supnrarrinfin
int
X
dX(x0 x)p dmicron(x) lt +infin
for some point x0 isin X
It is known that dP2 leW1 and that Wp leWq for any 1 le p le q
22 mm-Isomorphism and Lipschitz order
Definition 26 (mm-Space) Let (X dX) be a complete separable met-ric space and microX a Borel probability measure on X We call the triple(X dX microX) an mm-space We sometimes say that X is an mm-spacein which case the metric and the Borel measure of X are respectivelyindicated by dX and microX
Definition 27 (mm-Isomorphism) Two mm-spaces X and Y are saidto be mm-isomorphic to each other if there exists an isometry f supp microX rarr supp microY with flowastmicroX = microY where supp microX is the support ofmicroX Such an isometry f is called an mm-isomorphism Denote by Xthe set of mm-isomorphism classes of mm-spaces
Note that X is mm-isomorphic to (supp microX dX microX)We assume that an mm-space X satisfies
X = supp microX
unless otherwise stated
Definition 28 (Lipschitz order) Let X and Y be two mm-spaces Wesay that X (Lipschitz ) dominates Y and write Y ≺ X if there exists a1-Lipschitz map f X rarr Y satisfying flowastmicroX = microY We call the relation≺ on X the Lipschitz order
The Lipschitz order ≺ is a partial order relation on X
23 Observable diameter The observable diameter is one of themost fundamental invariants of an mm-space up to mm-isomorphism
Definition 29 (Partial and observable diameter) Let X be an mm-space and let κ gt 0 We define the κ-partial diameter diam(X 1minusκ) =diam(microX 1 minus κ) of X to be the infimum of the diameter of A whereA sub X runs over all Borel subsets with microX(A) ge 1 minus κ Denote byLip1(X) the set of 1-Lipschitz continuous real-valued functions on X We define the (κ-)observable diameter of X by
ObsDiam(X minusκ) = supfisinLip1(X)
diam(flowastmicroX 1minus κ)
ObsDiam(X) = infκgt0
maxObsDiam(X minusκ) κ
It is easy to see that the (κ-)observable diameter is monotone non-decreasing with respect to the Lipschitz order relation
8 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
24 Box distance and observable distance
Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I
It is known that any mm-space has a parameter
Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that
L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε
for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I
The box metric is a complete separable metric on X
Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that
(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x
prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε
We call the set X a nonexceptional domain of f
Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le
3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from
X to Y
Definition 214 (Observable distance) For any parameter ϕ of X weset
ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces
X and Y by
dconc(X Y ) = infϕψ
dH(ϕlowastLip1(X) ψlowastLip1(Y ))
where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X
It is known that dconc le and that the concentration topology isweaker than the box topology
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
4 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Gromov presents an exercise [9 31257] which is some easier special
cases of our theorem Our theorem provides not only an answer butalso a complete generalization of his exercise
For the case of round spheres (ie aij = a1j for all i and j) and alsoof projective spaces the theorem is formerly obtained by the secondnamed author [25 26] for which the convergence is only weak Alsothe weak convergence of Stiefel and flag manifolds are studied jointlyby Takatsu and the second named author [27]
We emphasize that convergence in the weakconcentration topol-ogy is completely different from weak convergence of measures Forinstance the Prokhorov distance between the normalized volume mea-sure on Snminus1(
radicnminus 1) and the n-dimensional standard Gaussian mea-
sure on Rn is bounded away from zero [27] though they both convergeto the infinite-dimensional standard Gaussian space in Gromovrsquos weaktopology
As for the characterization of weak convergence of measures weprove in Proposition 42 that if aiji l2-converges to an l2-sequence
ai as j rarr infin then the measure of En(j)
radicn(j)minus1 aiji
converges weakly to
the Gaussian measure γinfinai on a Hilbert space and consequently theweak convergence in Theorem 11 becomes the box convergence Con-versely the l2-convergence of aiji is also a necessary condition forthe box convergence of the (solid) ellipsoids as is seen in the followingtheorem
Theorem 12 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers satisfying (A0)ndash(A3) Then the convergence
in Theorem 11 becomes a box convergence if and only if we have
infinsum
i=1
a2i lt +infin and limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Theorems 11 and 12 together provide an example of irreduciblenontrivial convergent sequence of metric measure spaces in the con-centration topology ie the sequence of the (solid) ellipsoids with anl2-sequence ai and with a non-l2-convergent aiji as j rarr infin
The proof of the lsquoonly ifrsquo part of Theorem 12 is highly nontrivial Ifaiji does not l2-converge then it is easy to see that the measure ofthe (solid) ellipsoid in such the sequence does not converge weakly inthe Hilbert space However this is not enough to obtain the box non-convergence because we consider the isomorphism classes of (solid)ellipsoids for the box convergence For the complete proof we need adelicate discussion using Theorem 11
Let us briefly mention the outline of the proof of the weak con-vergence of solid ellipsoids in Theorem 11 For simplicity we set
HIGH-DIMENSIONAL ELLIPSOIDS 5
En = En(j)
radicn(j)minus1 aiji
and Γ = Γinfina2i
For the weak convergence it
is sufficient to show that
(11) the limit of En dominates Γ(12) Γ dominates the limit of En
where for two metric measure spaces X and Y the space X dominates
Y if there is a 1-Lipschitz map from X to Y preserving their measures(11) easily follows from the Maxwell-Boltzmann distribution law
(Proposition 32)(12) is much harder to prove Let us first consider the simple case
where En is the ball Bn(radicnminus 1) of radius
radicnminus 1 and where Γ = Γinfin
12
We see that for any fixed 0 lt θ lt 1 the n-dimensional Gaussian mea-sure γn12 and the normalized volume measure of Bn(θ
radicnminus 1) both are
very small for large n Ignoring this small part Bn(θradicnminus 1) we find
a measure-preserving isotropic map say ϕ from Γn12 Bn(θradicnminus 1)
to the annulus Bn(radicnminus 1) Bn(θ
radicnminus 1) where we normalize their
measures to be probability Estimating the Lipschitz constant of ϕwe obtain (12) with error This error is estimated and we eventuallyobtain the required weak convergence
We next try to apply this discussion for solid ellipsoids We considerthe distortion of the above isotropic map ϕ by a linear transforma-tion determined by aij However the Lipschitz constant of such thedistorted isotropic map is arbitrarily large depending on aij To over-come this problem we settle the assumptions (A0)ndash(A3) from whichthe discussion boils down to the special case where ai = aN for alli ge N and aij = ai ge aN for all i j and for a (large) number N Infact by (A0)ndash(A3) the solid ellipsoid En for large n and the Gaussianspace Γ are both close to those in the above special case In this specialcase the Gaussian measure γna2i
and the normalized volume measure
of En of the domain
x isin Rn o | |xi|x lt ε for any i = 1 N minus 1
are both almost full for large n and for any fixed ε gt 0 On this domainwe are able to estimate the Lipschitz constant of the distorted isotropicmap With some careful error estimates letting ǫ rarr 0+ and θ rarr 1minuswe prove the weak convergence of En to Γ
2 Preliminaries
In this section we survey the definitions and the facts needed in thispaper We refer to [9 Chapter 31
2+] and [25] for more details
21 Distance between measures
6 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Definition 21 (Total variation distance) The total variation distance
dTV(micro ν) of two Borel probability measures micro and ν on a topologicalspace X is defined by
dTV(micro ν) = supA
|micro(A)minus ν(A) |
where A runs over all Borel subsets of X
If micro and ν are both absolutely continuous with respect to a Borelmeasure ω on X then
dTV(micro ν) =1
2
int
X
∣∣∣∣dmicro
dωminus dν
dω
∣∣∣∣ dω
where dmicrodω
is the Radon-Nikodym derivative of micro with respect to ω
Definition 22 (Prokhorov distance) The Prokhorov distance dP(micro ν)between two Borel probability measures micro and ν on a metric space(X dX) is defined to be the infimum of ε ge 0 satisfying
micro(Bε(A)) ge ν(A)minus ε
for any Borel subset A sub X where Bε(A) = x isin X | dX(xA) lt ε The Prokhorov metric is a metrization of weak convergence of Borel
probability measures on X provided thatX is a separable metric spaceIt is known that dP le dTV
Definition 23 (Ky Fan distance) Let (X micro) be a measure space andY a metric space For two micro-measurable maps f g X rarr Y we definethe Ky Fan distance dKF(f g) between f and g to be the infimum ofε ge 0 satisfying
micro( x isin X | dY (f(x) g(x)) gt ε ) le ε
dKF is a pseudo-metric on the set of micro-measurable maps from X toY It holds that dKF(f g) = 0 if and only if f = g micro-ae We havedP(flowastmicro glowastmicro) le dKF(f g) where flowastmicro is the push-forward of micro by f
Let p be a real number with p ge 1 and (X dX) a complete separablemetric space
Definition 24 The p-Wasserstein distance between two Borel prob-ability measures micro and ν on X is defined to be
Wp(micro ν) = infπisinΠ(microν)
(int
XtimesXdX(x x
prime)p dπ(x xprime)
) 1p
(le +infin)
where Π(micro ν) is the set of couplings between micro and ν ie the set ofBorel probability measures π on X times X such that π(A times X) = micro(A)and π(X times A) = ν(A) for any Borel subset A sub X
Lemma 25 Let micro and micron n = 1 2 be Borel probability measures
on X Then the following (1) and (2) are equivalent to each other
(1) Wp(micron micro) rarr 0 as nrarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 7
(2) micron converges weakly to micro as nrarr infin and the p-th moment of micronis uniformly bounded
lim supnrarrinfin
int
X
dX(x0 x)p dmicron(x) lt +infin
for some point x0 isin X
It is known that dP2 leW1 and that Wp leWq for any 1 le p le q
22 mm-Isomorphism and Lipschitz order
Definition 26 (mm-Space) Let (X dX) be a complete separable met-ric space and microX a Borel probability measure on X We call the triple(X dX microX) an mm-space We sometimes say that X is an mm-spacein which case the metric and the Borel measure of X are respectivelyindicated by dX and microX
Definition 27 (mm-Isomorphism) Two mm-spaces X and Y are saidto be mm-isomorphic to each other if there exists an isometry f supp microX rarr supp microY with flowastmicroX = microY where supp microX is the support ofmicroX Such an isometry f is called an mm-isomorphism Denote by Xthe set of mm-isomorphism classes of mm-spaces
Note that X is mm-isomorphic to (supp microX dX microX)We assume that an mm-space X satisfies
X = supp microX
unless otherwise stated
Definition 28 (Lipschitz order) Let X and Y be two mm-spaces Wesay that X (Lipschitz ) dominates Y and write Y ≺ X if there exists a1-Lipschitz map f X rarr Y satisfying flowastmicroX = microY We call the relation≺ on X the Lipschitz order
The Lipschitz order ≺ is a partial order relation on X
23 Observable diameter The observable diameter is one of themost fundamental invariants of an mm-space up to mm-isomorphism
Definition 29 (Partial and observable diameter) Let X be an mm-space and let κ gt 0 We define the κ-partial diameter diam(X 1minusκ) =diam(microX 1 minus κ) of X to be the infimum of the diameter of A whereA sub X runs over all Borel subsets with microX(A) ge 1 minus κ Denote byLip1(X) the set of 1-Lipschitz continuous real-valued functions on X We define the (κ-)observable diameter of X by
ObsDiam(X minusκ) = supfisinLip1(X)
diam(flowastmicroX 1minus κ)
ObsDiam(X) = infκgt0
maxObsDiam(X minusκ) κ
It is easy to see that the (κ-)observable diameter is monotone non-decreasing with respect to the Lipschitz order relation
8 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
24 Box distance and observable distance
Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I
It is known that any mm-space has a parameter
Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that
L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε
for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I
The box metric is a complete separable metric on X
Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that
(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x
prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε
We call the set X a nonexceptional domain of f
Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le
3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from
X to Y
Definition 214 (Observable distance) For any parameter ϕ of X weset
ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces
X and Y by
dconc(X Y ) = infϕψ
dH(ϕlowastLip1(X) ψlowastLip1(Y ))
where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X
It is known that dconc le and that the concentration topology isweaker than the box topology
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 5
En = En(j)
radicn(j)minus1 aiji
and Γ = Γinfina2i
For the weak convergence it
is sufficient to show that
(11) the limit of En dominates Γ(12) Γ dominates the limit of En
where for two metric measure spaces X and Y the space X dominates
Y if there is a 1-Lipschitz map from X to Y preserving their measures(11) easily follows from the Maxwell-Boltzmann distribution law
(Proposition 32)(12) is much harder to prove Let us first consider the simple case
where En is the ball Bn(radicnminus 1) of radius
radicnminus 1 and where Γ = Γinfin
12
We see that for any fixed 0 lt θ lt 1 the n-dimensional Gaussian mea-sure γn12 and the normalized volume measure of Bn(θ
radicnminus 1) both are
very small for large n Ignoring this small part Bn(θradicnminus 1) we find
a measure-preserving isotropic map say ϕ from Γn12 Bn(θradicnminus 1)
to the annulus Bn(radicnminus 1) Bn(θ
radicnminus 1) where we normalize their
measures to be probability Estimating the Lipschitz constant of ϕwe obtain (12) with error This error is estimated and we eventuallyobtain the required weak convergence
We next try to apply this discussion for solid ellipsoids We considerthe distortion of the above isotropic map ϕ by a linear transforma-tion determined by aij However the Lipschitz constant of such thedistorted isotropic map is arbitrarily large depending on aij To over-come this problem we settle the assumptions (A0)ndash(A3) from whichthe discussion boils down to the special case where ai = aN for alli ge N and aij = ai ge aN for all i j and for a (large) number N Infact by (A0)ndash(A3) the solid ellipsoid En for large n and the Gaussianspace Γ are both close to those in the above special case In this specialcase the Gaussian measure γna2i
and the normalized volume measure
of En of the domain
x isin Rn o | |xi|x lt ε for any i = 1 N minus 1
are both almost full for large n and for any fixed ε gt 0 On this domainwe are able to estimate the Lipschitz constant of the distorted isotropicmap With some careful error estimates letting ǫ rarr 0+ and θ rarr 1minuswe prove the weak convergence of En to Γ
2 Preliminaries
In this section we survey the definitions and the facts needed in thispaper We refer to [9 Chapter 31
2+] and [25] for more details
21 Distance between measures
6 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Definition 21 (Total variation distance) The total variation distance
dTV(micro ν) of two Borel probability measures micro and ν on a topologicalspace X is defined by
dTV(micro ν) = supA
|micro(A)minus ν(A) |
where A runs over all Borel subsets of X
If micro and ν are both absolutely continuous with respect to a Borelmeasure ω on X then
dTV(micro ν) =1
2
int
X
∣∣∣∣dmicro
dωminus dν
dω
∣∣∣∣ dω
where dmicrodω
is the Radon-Nikodym derivative of micro with respect to ω
Definition 22 (Prokhorov distance) The Prokhorov distance dP(micro ν)between two Borel probability measures micro and ν on a metric space(X dX) is defined to be the infimum of ε ge 0 satisfying
micro(Bε(A)) ge ν(A)minus ε
for any Borel subset A sub X where Bε(A) = x isin X | dX(xA) lt ε The Prokhorov metric is a metrization of weak convergence of Borel
probability measures on X provided thatX is a separable metric spaceIt is known that dP le dTV
Definition 23 (Ky Fan distance) Let (X micro) be a measure space andY a metric space For two micro-measurable maps f g X rarr Y we definethe Ky Fan distance dKF(f g) between f and g to be the infimum ofε ge 0 satisfying
micro( x isin X | dY (f(x) g(x)) gt ε ) le ε
dKF is a pseudo-metric on the set of micro-measurable maps from X toY It holds that dKF(f g) = 0 if and only if f = g micro-ae We havedP(flowastmicro glowastmicro) le dKF(f g) where flowastmicro is the push-forward of micro by f
Let p be a real number with p ge 1 and (X dX) a complete separablemetric space
Definition 24 The p-Wasserstein distance between two Borel prob-ability measures micro and ν on X is defined to be
Wp(micro ν) = infπisinΠ(microν)
(int
XtimesXdX(x x
prime)p dπ(x xprime)
) 1p
(le +infin)
where Π(micro ν) is the set of couplings between micro and ν ie the set ofBorel probability measures π on X times X such that π(A times X) = micro(A)and π(X times A) = ν(A) for any Borel subset A sub X
Lemma 25 Let micro and micron n = 1 2 be Borel probability measures
on X Then the following (1) and (2) are equivalent to each other
(1) Wp(micron micro) rarr 0 as nrarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 7
(2) micron converges weakly to micro as nrarr infin and the p-th moment of micronis uniformly bounded
lim supnrarrinfin
int
X
dX(x0 x)p dmicron(x) lt +infin
for some point x0 isin X
It is known that dP2 leW1 and that Wp leWq for any 1 le p le q
22 mm-Isomorphism and Lipschitz order
Definition 26 (mm-Space) Let (X dX) be a complete separable met-ric space and microX a Borel probability measure on X We call the triple(X dX microX) an mm-space We sometimes say that X is an mm-spacein which case the metric and the Borel measure of X are respectivelyindicated by dX and microX
Definition 27 (mm-Isomorphism) Two mm-spaces X and Y are saidto be mm-isomorphic to each other if there exists an isometry f supp microX rarr supp microY with flowastmicroX = microY where supp microX is the support ofmicroX Such an isometry f is called an mm-isomorphism Denote by Xthe set of mm-isomorphism classes of mm-spaces
Note that X is mm-isomorphic to (supp microX dX microX)We assume that an mm-space X satisfies
X = supp microX
unless otherwise stated
Definition 28 (Lipschitz order) Let X and Y be two mm-spaces Wesay that X (Lipschitz ) dominates Y and write Y ≺ X if there exists a1-Lipschitz map f X rarr Y satisfying flowastmicroX = microY We call the relation≺ on X the Lipschitz order
The Lipschitz order ≺ is a partial order relation on X
23 Observable diameter The observable diameter is one of themost fundamental invariants of an mm-space up to mm-isomorphism
Definition 29 (Partial and observable diameter) Let X be an mm-space and let κ gt 0 We define the κ-partial diameter diam(X 1minusκ) =diam(microX 1 minus κ) of X to be the infimum of the diameter of A whereA sub X runs over all Borel subsets with microX(A) ge 1 minus κ Denote byLip1(X) the set of 1-Lipschitz continuous real-valued functions on X We define the (κ-)observable diameter of X by
ObsDiam(X minusκ) = supfisinLip1(X)
diam(flowastmicroX 1minus κ)
ObsDiam(X) = infκgt0
maxObsDiam(X minusκ) κ
It is easy to see that the (κ-)observable diameter is monotone non-decreasing with respect to the Lipschitz order relation
8 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
24 Box distance and observable distance
Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I
It is known that any mm-space has a parameter
Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that
L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε
for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I
The box metric is a complete separable metric on X
Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that
(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x
prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε
We call the set X a nonexceptional domain of f
Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le
3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from
X to Y
Definition 214 (Observable distance) For any parameter ϕ of X weset
ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces
X and Y by
dconc(X Y ) = infϕψ
dH(ϕlowastLip1(X) ψlowastLip1(Y ))
where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X
It is known that dconc le and that the concentration topology isweaker than the box topology
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
6 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Definition 21 (Total variation distance) The total variation distance
dTV(micro ν) of two Borel probability measures micro and ν on a topologicalspace X is defined by
dTV(micro ν) = supA
|micro(A)minus ν(A) |
where A runs over all Borel subsets of X
If micro and ν are both absolutely continuous with respect to a Borelmeasure ω on X then
dTV(micro ν) =1
2
int
X
∣∣∣∣dmicro
dωminus dν
dω
∣∣∣∣ dω
where dmicrodω
is the Radon-Nikodym derivative of micro with respect to ω
Definition 22 (Prokhorov distance) The Prokhorov distance dP(micro ν)between two Borel probability measures micro and ν on a metric space(X dX) is defined to be the infimum of ε ge 0 satisfying
micro(Bε(A)) ge ν(A)minus ε
for any Borel subset A sub X where Bε(A) = x isin X | dX(xA) lt ε The Prokhorov metric is a metrization of weak convergence of Borel
probability measures on X provided thatX is a separable metric spaceIt is known that dP le dTV
Definition 23 (Ky Fan distance) Let (X micro) be a measure space andY a metric space For two micro-measurable maps f g X rarr Y we definethe Ky Fan distance dKF(f g) between f and g to be the infimum ofε ge 0 satisfying
micro( x isin X | dY (f(x) g(x)) gt ε ) le ε
dKF is a pseudo-metric on the set of micro-measurable maps from X toY It holds that dKF(f g) = 0 if and only if f = g micro-ae We havedP(flowastmicro glowastmicro) le dKF(f g) where flowastmicro is the push-forward of micro by f
Let p be a real number with p ge 1 and (X dX) a complete separablemetric space
Definition 24 The p-Wasserstein distance between two Borel prob-ability measures micro and ν on X is defined to be
Wp(micro ν) = infπisinΠ(microν)
(int
XtimesXdX(x x
prime)p dπ(x xprime)
) 1p
(le +infin)
where Π(micro ν) is the set of couplings between micro and ν ie the set ofBorel probability measures π on X times X such that π(A times X) = micro(A)and π(X times A) = ν(A) for any Borel subset A sub X
Lemma 25 Let micro and micron n = 1 2 be Borel probability measures
on X Then the following (1) and (2) are equivalent to each other
(1) Wp(micron micro) rarr 0 as nrarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 7
(2) micron converges weakly to micro as nrarr infin and the p-th moment of micronis uniformly bounded
lim supnrarrinfin
int
X
dX(x0 x)p dmicron(x) lt +infin
for some point x0 isin X
It is known that dP2 leW1 and that Wp leWq for any 1 le p le q
22 mm-Isomorphism and Lipschitz order
Definition 26 (mm-Space) Let (X dX) be a complete separable met-ric space and microX a Borel probability measure on X We call the triple(X dX microX) an mm-space We sometimes say that X is an mm-spacein which case the metric and the Borel measure of X are respectivelyindicated by dX and microX
Definition 27 (mm-Isomorphism) Two mm-spaces X and Y are saidto be mm-isomorphic to each other if there exists an isometry f supp microX rarr supp microY with flowastmicroX = microY where supp microX is the support ofmicroX Such an isometry f is called an mm-isomorphism Denote by Xthe set of mm-isomorphism classes of mm-spaces
Note that X is mm-isomorphic to (supp microX dX microX)We assume that an mm-space X satisfies
X = supp microX
unless otherwise stated
Definition 28 (Lipschitz order) Let X and Y be two mm-spaces Wesay that X (Lipschitz ) dominates Y and write Y ≺ X if there exists a1-Lipschitz map f X rarr Y satisfying flowastmicroX = microY We call the relation≺ on X the Lipschitz order
The Lipschitz order ≺ is a partial order relation on X
23 Observable diameter The observable diameter is one of themost fundamental invariants of an mm-space up to mm-isomorphism
Definition 29 (Partial and observable diameter) Let X be an mm-space and let κ gt 0 We define the κ-partial diameter diam(X 1minusκ) =diam(microX 1 minus κ) of X to be the infimum of the diameter of A whereA sub X runs over all Borel subsets with microX(A) ge 1 minus κ Denote byLip1(X) the set of 1-Lipschitz continuous real-valued functions on X We define the (κ-)observable diameter of X by
ObsDiam(X minusκ) = supfisinLip1(X)
diam(flowastmicroX 1minus κ)
ObsDiam(X) = infκgt0
maxObsDiam(X minusκ) κ
It is easy to see that the (κ-)observable diameter is monotone non-decreasing with respect to the Lipschitz order relation
8 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
24 Box distance and observable distance
Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I
It is known that any mm-space has a parameter
Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that
L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε
for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I
The box metric is a complete separable metric on X
Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that
(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x
prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε
We call the set X a nonexceptional domain of f
Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le
3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from
X to Y
Definition 214 (Observable distance) For any parameter ϕ of X weset
ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces
X and Y by
dconc(X Y ) = infϕψ
dH(ϕlowastLip1(X) ψlowastLip1(Y ))
where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X
It is known that dconc le and that the concentration topology isweaker than the box topology
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
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[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
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Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
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[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
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[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
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[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
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[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
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[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
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[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
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[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 7
(2) micron converges weakly to micro as nrarr infin and the p-th moment of micronis uniformly bounded
lim supnrarrinfin
int
X
dX(x0 x)p dmicron(x) lt +infin
for some point x0 isin X
It is known that dP2 leW1 and that Wp leWq for any 1 le p le q
22 mm-Isomorphism and Lipschitz order
Definition 26 (mm-Space) Let (X dX) be a complete separable met-ric space and microX a Borel probability measure on X We call the triple(X dX microX) an mm-space We sometimes say that X is an mm-spacein which case the metric and the Borel measure of X are respectivelyindicated by dX and microX
Definition 27 (mm-Isomorphism) Two mm-spaces X and Y are saidto be mm-isomorphic to each other if there exists an isometry f supp microX rarr supp microY with flowastmicroX = microY where supp microX is the support ofmicroX Such an isometry f is called an mm-isomorphism Denote by Xthe set of mm-isomorphism classes of mm-spaces
Note that X is mm-isomorphic to (supp microX dX microX)We assume that an mm-space X satisfies
X = supp microX
unless otherwise stated
Definition 28 (Lipschitz order) Let X and Y be two mm-spaces Wesay that X (Lipschitz ) dominates Y and write Y ≺ X if there exists a1-Lipschitz map f X rarr Y satisfying flowastmicroX = microY We call the relation≺ on X the Lipschitz order
The Lipschitz order ≺ is a partial order relation on X
23 Observable diameter The observable diameter is one of themost fundamental invariants of an mm-space up to mm-isomorphism
Definition 29 (Partial and observable diameter) Let X be an mm-space and let κ gt 0 We define the κ-partial diameter diam(X 1minusκ) =diam(microX 1 minus κ) of X to be the infimum of the diameter of A whereA sub X runs over all Borel subsets with microX(A) ge 1 minus κ Denote byLip1(X) the set of 1-Lipschitz continuous real-valued functions on X We define the (κ-)observable diameter of X by
ObsDiam(X minusκ) = supfisinLip1(X)
diam(flowastmicroX 1minus κ)
ObsDiam(X) = infκgt0
maxObsDiam(X minusκ) κ
It is easy to see that the (κ-)observable diameter is monotone non-decreasing with respect to the Lipschitz order relation
8 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
24 Box distance and observable distance
Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I
It is known that any mm-space has a parameter
Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that
L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε
for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I
The box metric is a complete separable metric on X
Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that
(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x
prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε
We call the set X a nonexceptional domain of f
Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le
3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from
X to Y
Definition 214 (Observable distance) For any parameter ϕ of X weset
ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces
X and Y by
dconc(X Y ) = infϕψ
dH(ϕlowastLip1(X) ψlowastLip1(Y ))
where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X
It is known that dconc le and that the concentration topology isweaker than the box topology
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
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[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
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HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
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[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
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[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
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order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
8 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
24 Box distance and observable distance
Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I
It is known that any mm-space has a parameter
Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that
L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε
for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I
The box metric is a complete separable metric on X
Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that
(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x
prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε
We call the set X a nonexceptional domain of f
Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le
3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from
X to Y
Definition 214 (Observable distance) For any parameter ϕ of X weset
ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces
X and Y by
dconc(X Y ) = infϕψ
dH(ϕlowastLip1(X) ψlowastLip1(Y ))
where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X
It is known that dconc le and that the concentration topology isweaker than the box topology
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
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[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
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[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
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[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
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[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
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[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
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[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
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transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
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order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
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ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 9
25 Pyramid
Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)
(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space
Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed
We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π
For an mm-space X we define
PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X
We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid
We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly
to a pyramid if it converges with respect to ρ We have the following
(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ
(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π
In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1
For an mm-space X a pyramid P and t gt 0 we define
tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ
We have the following
Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have
ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))
le 2 dP(micro ν) le 2 dTV(micro ν)
26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
10 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proposition 217 The sequence Xn infinitely dissipates if and only
if PXn converges weakly to X as nrarr infin
An easy discussion using [19 Lemma 66] leads to the following
Proposition 218 The following (1) and (2) are equivalent to each
other
(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-
ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates
27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant
Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-
alent to each other
(1) P is concentrated
(2) There exists a sequence of mm-spaces asymptotically concen-
trating to P
(3) If a sequence of mm-spaces converges weakly to P then it asymp-
totically concentrates
28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product
γna2i =
notimes
i=1
γ1a2i
of the one-dimensional centered Gaussian measure γ1a2i
of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i
is possibly degenerate We
call the mm-space Γna2i = (Rn middot γna2
i) the n-dimensional Gaussian
space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i
where a2i are
the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative
real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie
πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn
Since the projection πnnminus1 Γna2i rarr Γnminus1
a2i is 1-Lipschitz continuous
and measure-preserving for any n ge 2 the Gaussian space Γna2i is
monotone nondecreasing in n with respect to the Lipschitz order so
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 11
that as n rarr infin the associated pyramid PΓna2i converges weakly to
the -closure of⋃infinn=1PΓna2i
denoted by PΓinfina2i
We call PΓinfina2i
the
virtual Gaussian space with variance a2i We remark that the infiniteproduct measure
γinfina2i =
infinotimes
i=1
γ1a2i
is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where
(21)infinsum
i=1
a2i lt +infin
the measure γinfina2i is a Borel measure with respect to the l2-norm middot
which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin
a2i = (H middot γinfina2i ) is
an mm-space In the case of (21) the variance of γinfina2i satisfies
int
Rn
x2 dγinfina2i (x) =infinsum
i=1
a2i
3 Weak convergence of ellipsoids
In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem
Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R
n rarr Rn defined by
Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn
We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-
malized Lebesgue measure ǫnαi = Ln|Enαi
where micro = micro(X)minus1micro is
the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean
distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ
nminus1 of
the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En
αi enαi) be any one of
(Enαi ǫnαi) and (Snminus1
αi σnminus1αi)
for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En
αi
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
12 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Lemma 31 Let αi and βi i = 1 2 n be two sequences of
positive real numbers If αi le βi for all i = 1 2 n then Enαi is
dominated by Enβi
Proof The map Lnαiβi Enβi rarr En
αi is 1-Lipschitz continuous andpreserves their measures
Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers
satisfying (A0) and (A3) Then (πn(j)k )lowaste
n
radicn(j)minus1 aiji
converges weakly
to γkai as j rarr infin for any fixed positive integer k where πnk is defined
in Subsection 28
Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])
Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of positive real numbers If supi aij diverges to infinity as
j rarr infin then En(j)
radicn(j)minus1 aiji
infinitely dissipates
Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-
Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution
law (Proposition 32) we have
lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge lim infjrarrinfin
ObsDiam(En(j)
radicn(j)minus1 aiji
minusκ)
ge limjrarrinfin
diam((πn(j)1 )lowaste
n(j)
radicn(j)minus1 aiji
1minus κ)
= diam(γ1a 1minus κ)
which diverges to infinity as a rarr infin Proposition 218 leads us to the
dissipation property for En(j)
radicn(j)minus1 aiji
Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i
to ǫnradicnminus1 ai For r ge 0 we
determine a real number R = R(r) in such a way that 0 le R leradicnminus 1
and γn12(Br(o)) = ǫnradicnminus1
(BR(o)) where ǫnradicnminus1
denotes the normalized
Lebesgue measure on Enradicnminus1
= Bradicnminus1(o) sub Rn Define an isotropic
map ϕ Rn rarr Enradicnminus1
by
ϕ(x) =R(x)x x x isin Rn
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 13
We remark that ϕlowastγn12 = ǫnradic
nminus1 It holds that
R = (nminus 1)12
(1
Inminus1
int r
0
tnminus1eminust2
2 dt
) 1n
where
Im =
int infin
0
tmeminust2
2 dt
Note that R is strictly monotone increasing in r Let L = Lnai and
r = r(x) = Lminus1(x) We define
ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai
The map ϕE is a transport map from γna2i to ǫn
radicnminus1 ai ie ϕ
Elowastγ
na2i
=
ǫnradicnminus1 ai It holds that ϕE(x) = R
rx if x 6= o We denote by ϕS
Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie
ϕS(x) =
radicnminus 1
rx x isin Rn o
which is a transport map from γna2i to σnminus1
radicnminus1ai
For an integer N with 1 le N le n and for ε gt 0 we define
DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1
For 0 lt θ lt 1 let
F nθ = x isin Rn | Lminus1(x) ge θ
radicn
Lemma 34 We assume that
(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N
with N le n
Then there exists a universal positive real number C such that for
any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the
operator norms of the differentials of ϕE and ϕS satisfy
dϕEx le
radic1 + CNε
θand dϕS
x leradic1 + CNε
θ
for any x isin DnNε cap F n
θ
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
14 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
Proof Let x isin DnNε capF n
θ be any point We first estimate dϕEx Take
any unit vector v isin Rn We see that
dϕEx(v)2 =
nsum
j=1
(part
partr
(R
r
)partr
partxj〈x v〉+ R
rvj
)2
=1
r2
(part
partr
(R
r
))2
〈x v〉2nsum
j=1
x2ja4j
+ 2R
r2part
partr
(R
r
)〈x v〉
nsum
j=1
vjxja2j
+R2
r2
It follows from (i) (ii) and x isin DnNε that
a2r2
x2 = 1 +Nminus1sum
j=1
(a2
a2jminus 1
)x2j
x2 = 1 +O(Nε2)
and so
ar
x = 1 +O(Nε2)xar
= 1 +O(Nε2)
We also have
a4
x2nsum
j=1
x2ja4j
= 1 +Nminus1sum
j=1
(a4
a4jminus 1
)x2j
x2 = 1 + O(Nε2)
a2
x
nsum
j=1
vjxja2j
=
nsum
j=1
vjxjx +
Nminus1sum
j=1
(a2
a2jminus 1
)vjxjx =
〈x v〉x +O(Nε)
By these formulas setting t = 〈x v〉x and g = r partpartr
(Rr
) we have
dϕEx(v)2 = t2g2(1 +O(Nε2)) +
2t2Rg
r(1 +O(Nε2))(31)
+2tRg
rO(Nε) +
R2
r2
We are going to estimate g Letting f(r) =int r0tnminus1eminus
t2
2 dt we have
partR
partr=
radicnminus 1nminus1I
minus 1n
nminus1f(r)1nminus1rnminus1eminus
r2
2 le nminus 12f(r)minus1rnminus1eminus
r2
2
which together with f(r) geint r0tnminus1 dt = rn
nand r ge θ
radicn yields
0 le partR
partrle
radicn
rle 1
θ
Since R leradicnminus 1 and r ge θ
radicn we have 0 le Rr lt 1θ Therefore
|g| =∣∣∣∣partR
partrminus R
r
∣∣∣∣ le1
θ
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 15
Thus (31) is reduced to
dϕEx(v)2 = t2g2 +
2t2Rg
r+R2
r2+O(θminus2Nε)
= t2(partR
partr
)2
+ (1minus t2)R2
r2+O(θminus2Nε)
le θminus2 +O(θminus2Nε)
This completes the required estimate of dϕEx(v)
If we replace R withradicnminus 1 then ϕE becomes ϕS and the above
formulas are all true also for ϕS This completes the proof
We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions
(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N
Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have
limnrarrinfin
enradicnminus1ai(D
nNε) = 1(1)
limnrarrinfin
γna2i (Dn
Nε cap F nθ ) = 1(2)
Proof Lemma [25 Lemma 741] tells us that γna2i (F n
θ ) = γn12(Lminus1(F n
θ ))
tends to 1 as nrarr infin It holds that enradicnminus1 ai(D
nNε) = σnminus1
radicnminus1 ai
(DnNε) =
γna2i (Dn
Nε) The Maxwell-Boltzmann distribution law leads us that
σnminus1radicnminus1 ai
(DnNε) converges to 1 as n rarr infin This completes the proof
Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain
converges weakly to a pyramid Pinfin as nrarr infin then
Pinfin sub PΓinfina2i
Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1
radic1 + CNε where C is the constant in Lemma 34 Note that
θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let
ϕ =
ϕE if (En
radicnminus1ai e
nradicnminus1ai) = (Enradicnminus1 ai ǫ
nradicnminus1ai)
ϕS if (Enradicnminus1ai e
nradicnminus1ai) = (Snminus1
radicnminus1 ai σ
nminus1radicnminus1 ai)
Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n
θ and since ϕlowast( ˜γna2i |Dn
NεcapFn
θ) =
˜enradicnminus1 ai|Dn
Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot
˜enradicnminus1ai|Dn
Nε) is dominated by Yn = (Rn middot ˜γna2i
|DnNε
capFnθ) and so
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
16 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin
ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en
radicnminus1ai|Dn
Nε en
radicnminus1 ai) rarr 0
ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i
|DnNε
capFnθ γna2i
) rarr 0
Therefore θ2Pinfin is contained in PΓinfina2i
As ε rarr 0+ we have θ rarr 1
and θ2Pinfin rarr Pinfin This completes the proof
Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to
PΓinfina2i
as nrarr infin
Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge
weakly to PΓinfina2i
as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)
radicn(j)minus1 ai
converges weakly to a pyramid Pinfin
different from PΓinfina2i
The Maxwell-Boltzmann distribution law tells us that the push-
forward measure νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 ai
converges weakly to γkai
as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i
Since En(j)
radicn(j)minus1 ai
dominates (Rk middot νkn(j)) the limit pyramid Pinfin
contains Γka2i for any k This proves
(32) Pinfin sup PΓinfina2i
Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is
dominated by Enradicnminus1 ai which implies PEn
radicnminus1 ai sub PEn
radicnminus1 ai for
any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn
radicnminus1 ain is contained in PΓinfin
a2i Therefore Pinfin is
contained in PΓinfina2i
Denote by l the number of irsquos with ai lt a For
any k ge N we consider the projection from Γk+la2i to Γka2i
dropping
the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2
i dominates Γka2i
and so PΓinfina2i
supPΓinfin
a2i We thus obtain
(33) Pinfin sub PΓinfina2i
Combining (32) and (33) yields Pinfin = PΓinfina2i
which is a contra-
diction This completes the proof
Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sup PΓinfina2i
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 17
Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-
Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste
n(j)
radicn(j)minus1 aiji
converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0
The ellipsoid En(j)
radicn(j)minus1 aiji
dominates (Rk middot νkn(j)) which converges
to Γka2i so that Γka2i
belongs to Pinfin Since Γka2i for any k ge i0 is
mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma
Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1
radicn(j)minus1 aiji
converges
weakly to a pyramid Pinfin as j rarr infin then
Pinfin sub PΓinfina2i
Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin
We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that
(34) ai le (1 + ε)ainfin for any i ge I(ε)
Also there is a number J(ε) such that
(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)
By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)
It follows from (35) and ainfin le ai that
(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)
Let
bεi =
ai if i le I(ε)
ainfin if i gt I(ε)
By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E
n(j)aiji ≺ E
n(j)(1+2ε)bεi = (1 + 2ε)E
n(j)bεi Lemma 37 implies that
PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin
is contained in (1 + 2ε)PΓinfinb2
εi for any ε gt 0 Since bεi le ai we see
that Pinfin is contained in (1 + 2ε)PΓinfina2i
for any ε gt 0 This proves the
lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such
that
(38) ai lt ε for any i ge I(ε)
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
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[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
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[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
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[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
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order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
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28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
18 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that
aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)
aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)
It follows from (38) and (39) that
(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)
Let
bεi =
(1 + ε)ai if i lt I(ε)
2ε if i ge I(ε)
From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so
En(j)aiji ≺ E
n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)
radicn(j)minus1 bεi
converges weakly to PΓinfinb2εi
as j rarr infin Therefore Pinfin is contained
in PΓinfinb2εi
for any ε gt 0 Let k be any number with k ge I(ε) The
Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ
I(ε)minus1
(1+ε)2a2i
and ΓkminusI(ε)+1
(2ε)2 It follows from the Gaussian isoperimetry that
ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf
κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)
which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]
ρ(PΓkb2εiPΓ
I(ε)minus1
(1+ε)2a2i ) le dconc(Γ
kb2εi
ΓI(ε)minus1
(1+ε)2a2i ) le τ(ε)
Taking the limit as k rarr infin yields
ρ(PΓinfinb2εi
PΓI(ε)minus1
(1+ε)2a2i) le τ(ε)
There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin
b2ε(l)i
converges weakly to a pyramid P primeinfin as l rarr
infin P primeinfin contains Pinfin and PΓ
I(ε(l))minus1
(1+ε(l))2a2i converges weakly to P prime
infin as
l rarr infin Since PΓI(ε(l))minus1
(1+ε(l))2a2i is contained in PΓinfin
(1+ε(l))2a2i and since
PΓinfin(1+ε(l))2a2i
= (1+ε(l))PΓinfina2i
converges weakly to PΓinfina2i
as l rarr infin
the pyramid P primeinfin is contained in PΓinfin
a2i so that Pinfin is contained in
PΓinfina2i
This completes the proof
Proof of Theorem 11 Suppose that PEn(j)
radicn(j)minus1 aiji
does not converge
weakly to PΓinfina2i
as j rarr infin Then taking a subsequence of j we
may assume that PEn(j)
radicn(j)minus1 aiji
converges weakly to a pyramid Pinfin
different from PΓinfina2i
which contradicts Lemmas 38 and 39 Thus
PEn(j)
radicn(j)minus1 aiji
converges weakly to PΓinfina2
i as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 19
As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin
a2i is well-defined as an mm-space if and only if ai is an
l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology
Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1
a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i
asymptotically
(spectrally) concentrates to Γinfinai
Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin
a2 is not
concentrated Since PΓinfina2
i contains PΓinfin
a2 the pyramid PΓinfina2
i is not
concentrated which implies that En(j)
radicn(j)minus1 aiji
does not asymptotically
concentrate (see Theorem 219)This completes the proof of the theorem
Let us next consider the convergence of the Gaussian spaces
Proposition 310 Let aij i = 1 2 n(j) j = 1 2 be a
sequence of nonnegative real numbers If supi aij diverges to infinity as
j rarr infin then Γn(j)
a2ijinfinitely dissipates
Proof Exchanging the coordinates we assume that a1j diverges to in-
finity as j rarr infin Since Γ1a21j
is dominated by Γn(j)
a2ij we have
ObsDiam(Γn(j)
a2ijminusκ) ge diam(Γ1
a21j 1minus κ) rarr infin as j rarr infin
This together with Proposition 218 completes the proof
In a similar way as in the proof of Theorem 11 we obtain the fol-lowing
Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)
a2ijconverges
weakly to PΓinfina2i
as j rarr infin This convergence becomes a convergence in
the concentration topology if and only if ai is an l2-sequence More-
over this convergence becomes an asymptotic concentration if and only
if ai converges to zero
Proof Suppose that Γn(j)
a2ijdoes not converge weakly to PΓinfin
a2i as j rarr
infin Then there is a subsequence of PΓn(j)
a2ijj that converges weakly
to a pyramid Pinfin different from PΓinfina2i
We write such a subsequence
by the same notation PΓn(j)
a2ijj
Since Γka2ijis dominated by Γ
n(j)
a2ijfor k le n(j) and Γka2ij
converges
weakly to Γka2i as j rarr infin we see that Γka2i
belongs to Pinfin for any k
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
20 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
so that
Pinfin sup PΓinfina2i
We prove Pinfin sub PΓinfina2i
in the case of ainfin gt 0 where ainfin = limirarrinfin ai
Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)
Γka2ijis dominated by (1 + ε)Γka2i
and so PΓn(j)
a2ijisub (1 + ε)PΓinfin
a2i
This proves Pinfin sub PΓinfina2i
We next prove Pinfin sub PΓinfina2i
in the case of ainfin = 0 Let bεi be as
in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin
b2εi We obtain Pinfin sub PΓinfin
a2i in the same way as in the
proof of Lemma 39 The weak convergence of Γn(j)
a2ijto PΓinfin
b2εi
has
been provedThe rest is identical to the proof of Theorem 11 This completes the
proof
4 Box convergence of ellipsoids
The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en
radicnminus1aij if aij l2-
converges
Lemma 41 Let A be a family of sequences of positive real numbers
such that
supaiisinA
infinsum
i=1
a2i lt +infin
Then we have
limnrarrinfin
supaiisinA
dP(ǫnradicnminus1 ai γ
na2i
) = 0(1)
lim supnrarrinfin
supaiisinA
W2(σnminus1radicnminus1 ai γ
na2i
)2 leradic2eminus1 sup
aiisinA
infinsum
i=k+1
a2i(2)
for any positive integer k
Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn
radicnminus1 ai|rminus1([ θ
radicnminus1
radicnminus1 ]) and γ
na2i
|rminus1([ θradicnminus1θminus1
radicnminus1 ])
which we denote by ǫnθ and γnθ respectively Set
vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le
radicnminus 1 )
wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1
radicnminus 1 )
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 21
We remark that
vθn = ǫnradicnminus1 ai(r
minus1([ θradicnminus 1
radicnminus 1 ]))
wθn = γna2i (rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ]))
limnrarrinfin
vθn = limnrarrinfin
wθn = 1
It then holds that
dP(ǫnradicnminus1ai ǫ
nθ ) le dTV(ǫ
nradicnminus1 ai ǫ
nθ ) = 1minus vθn(41)
dP(γna2i
γnθ ) le dTV(γna2i
γnθ ) = 1minus wθn(42)
To estimate dP(ǫnθ γ
nθ ) we define a transport map say ψ from γnθ to
ǫnθ in the same manner as for ϕE in Section 3 which is expressed as
ψ(x) =R
rx x isin rminus1([ θ
radicnminus 1 θminus1
radicnminus 1 ])
where R is the function of variable r isin [ θradicnminus 1 θminus1
radicnminus 1 ] defined
by
θradicnminus 1 le R le
radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))
Since θ2 le Rr le θminus1 we have
W2(ǫnθ γ
nθ )
2 leint
Rn
ψ(x)minus x 2 dγnθ (x) =int
Rn
(R
rminus 1
)2
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2int
Rn
x2 dγnθ (x)
le max(1minus θ)2 (θ2 minus 1)2γna2i
(rminus1([ θradicnminus 1 θminus1
radicnminus 1 ]))
int
Rn
x2 dγna2i (x)
=max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
which together with (41) and (42) implies
dP(ǫnradicnminus1ai γ
na2i
)
le 2minus vθn minus wθn +
(max(1minus θ)2 (θ2 minus 1)2
wθn
nsum
i=1
a2i
) 14
and hence
lim supnrarrinfin
supaiisinA
dP(ǫnradicnminus1ai γ
na2i
)
le(max(1minus θ)2 (θ2 minus 1)2 sup
aiisinA
infinsum
i=1
a2i
) 14
rarr 0 as θ rarr 1+
This proves (1)
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
22 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We prove (2) Using the transport map ϕS from γna2i to σnminus1
radicnminus1ai
in Section 3 we have
W2(σnminus1radicnminus1 ai γ
na2i
)2 leint
Rn
∥∥ z minus ϕS(z)∥∥2 dγna2i (z)
=
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x) middot 1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr
where Im =intinfin0tmeminust
22 dt We see in the proof of [25 Lemma 741]
that rmeminusr22 le mm2eminusm2eminus(rminusradic
m)22 and also that Im sim radicπ(m minus
1)m2eminus(mminus1)2 Therefore
1
Inminus1
int infin
0
(r minusradicnminus 1)2rnminus1eminusr
22 dr le 1
Inminus1
radic2π(nminus 1)(nminus1)2eminus(nminus1)2
simradic2eminus12
(1minus 1(nminus 1))(nminus1)2minusrarr
radic2eminus1 as nrarr infin
For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |
|xi| lt ε for i = 1 2 k Thenint
Snminus1(1)Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )
nsum
i=1
a2i
int
Snminus1kε
nsum
i=1
a2ix2i dσ
nminus1(x) le ε2ksum
i=1
a2i +nsum
i=k+1
a2i
which imply
supaiisinA
int
Snminus1(1)
nsum
i=1
a2ix2i dσ
nminus1(x)
le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup
aiisinA
infinsum
i=1
a2i + supaiisinA
infinsum
i=k+1
a2i
Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)
This completes the proof
Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-
quence of positive real numbers where n(j) j = 1 2 is a se-
quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume
limjrarrinfin
n(j)sum
i=1
(aij minus ai)2 = 0
Then we have
(1) ǫn(j)
radicn(j)minus1 aiji
converges weakly to γinfina2ii as j rarr infin
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 23
(2) σn(j)minus1
radicn(j)minus1 aiji
converges to γinfina2i iin the 2-Wasserstein metric as
j rarr infin
In particular En(j)
radicn(j)minus1 aiji
box converges to Γinfina2i i
as j rarr infin
Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies
lim supjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 aiji
γn(j)
a2iji)
2
leradic2eminus1 lim sup
jrarrinfin
infinsum
i=k+1
a2ij =radic2eminus1
infinsum
i=k+1
a2i minusrarr 0 as k rarr infin
Gelbrichrsquos formula [7] tells us that
W2(γn(j)
a2iji γinfina2i
)2 =infinsum
i=1
(aij minus ai)2 minusrarr 0 as j rarr infin
By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-
ing dP2 leW2 This completes the proof
Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of
positive real numbers where n(j) j = 1 2 is a sequence of pos-
itive integers divergent to infinity Ifsumn(j)
i=1 b2ij converges to a positive
real number as j rarr infin then en(j)radicn(j)minus1 bij
has no subsequence con-
verging weakly to the Dirac measure δo at the origin o in H where we
embed the (solid) ellipsoids En(j)
radicn(j)minus1 bij
sub Rn(j) into the Hilbert space
H naturally and consider en(j)
radicn(j)minus1 biji
as Borel probability measures
on H
Proof We first prove the lemma for σn(j)minus1
radicn(j)minus1 bij
It holds that
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 =
int
Sn(j)minus1
radic
n(j)minus1 biji
y2 dσn(j)minus1
radicn(j)minus1 biji
(y)
=
int
Snminus1(1)
n(j)sum
i=1
(n(j)minus 1)b2ijx2i dσ
n(j)minus1(x)
=
n(j)sum
i=1
b2ij
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x)
It follows from the Maxwell-Boltzmann distribution law that
limjrarrinfin
(n(j)minus 1)
int
Snminus1(1)
x21 dσn(j)minus1(x) = 1
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
24 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
We therefore have
limjrarrinfin
W2(σn(j)minus1
radicn(j)minus1 biji
δo)2 = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
so that σn(j)minus1
radicn(j)minus1 biji
has no subsequence converging weakly to δo
We prove the lemma for ǫn(j)
radicn(j)minus1 bij
Applying Lemma 41(1) yields
thatlimjrarrinfin
dP(ǫn(j)
radicn(j)minus1 bij
γn(j)
b2ij) = 0
We also have
limjrarrinfin
W2(γn(j)
b2ij δo)
2 = limjrarrinfin
int
Rn
x2 dγn(j)b2ij(x) = lim
jrarrinfin
n(j)sum
i=1
b2ij gt 0
and hence γn(j)b2ij does not have a subsequence converging weakly to δo
and so does ǫn(j)radicn(j)minus1 bij
This completes the proof of the lemma
The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space
Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence
of positive real numbers where n(j) j = 1 2 is a sequence of
positive integers divergent to infinity We assume that
limjrarrinfin
aij = 0 for any i(i)
lim infjrarrinfin
n(j)sum
i=1
a2ij gt 0(ii)
Then there exists no box convergent subsequence of En(j)
radicn(j)minus1 aiji
Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-
tone nonincreasing in i for each j We suppose that En(j)
radicn(j)minus1 aiji
has a box convergent subsequence for which we use the same nota-
tion Then by (i) and Theorem 11 the box limit of En(j)
radicn(j)minus1 aiji
is mm-isomorphic to a one-point mm-space We set
Aj =
n(j)sum
i=1
a2ij
12
and bij =aij
maxAj 1
Since bij le aij we see that En(j)
radicn(j)minus1 biji
is dominated by En(j)
radicn(j)minus1 aiji
so that En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space as j rarr
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 25
infin We remark that
lim infjrarrinfin
n(j)sum
i=1
b2ij gt 0 and
n(j)sum
i=1
b2ij le 1
Taking a subsequence again we assume thatsumn(j)
i=1 b2ij converges to a
positive real number as j rarr infin Applying Lemma 43 yields that
en(j)radicn(j)minus1 bij
has no subsequence converging weakly to δo in H Since
En(j)
radicn(j)minus1 biji
box converges to a one-point mm-space say lowast as j rarr infin
Lemma 213 implies that there is a sequence of εj-mm-isomorphisms
fj En(j)
radicn(j)minus1 biji
rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional
domain of fj has en(j)
radicn(j)minus1 bij
-measure at least 1minus εj and diameter at
most εj There is a closed metric ball Bj sub H of radius εj that contains
the nonexceptional domain of fj Note that en(j)
radicn(j)minus1 biji
(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely
many j then a subsequence of en(j)radicn(j)minus1 biji
would converge weakly
to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj
Since en(j)
radicn(j)minus1 biji
is centrally symmetric with respect to the origin
we see that en(j)
radicn(j)minus1 aiji
(minusBj) = en(j)
radicn(j)minus1 aiji
(Bj) ge 1minus εj which is
a contradiction if j is large enough This completes the proof
Lemma 45 Let Xn n = 1 2 be a box convergent sequence
of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with
Yn ≺ Xn Then Yn has a box convergent subsequence
Proof The lemma follows from [25 Lemma 428]
Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42
We prove the lsquoonly ifrsquo part Suppose that En(j)
radicn(j)minus1 aiji
is box
convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not
then by Theorem 11 the weak limit of En(j)
radicn(j)minus1 aiji
is not an
mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that
limjrarrinfinsumn(j)
i=1 a2ij exists in [ 0+infin ] We prove
(43) limjrarrinfin
n(j)sum
i=1
a2ij gt
infinsum
i=1
a2i
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
26 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)
Take a real number ε0 in such a way that
0 lt ε0 lt limjrarrinfin
n(j)sum
i=1
a2ij minusinfinsum
i=1
a2i
Setting
aijk =
akj if i le k
aij if i ge k + 1
we have
limjrarrinfin
n(j)sum
i=1
a2ijk = limjrarrinfin
ksum
i=1
a2ijk +
n(j)sum
i=k+1
a2ijk
= ka2k + limjrarrinfin
n(j)sum
i=k+1
a2ij
= ka2k + limjrarrinfin
n(j)sum
i=1
a2ij minusksum
i=1
a2i gt ε0
Thus for any positive integer k there is j(k) such that
n(j(k))sum
i=1
a2ij(k)k gt ε0 and | akj(k) minus ak | lt1
k
Letting bik = aij(k)k we observe the following
bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin
bullsumn(j(k))
i=1 b2ik gt ε0 gt 0 for any k
Consider Ek = En(j(k))
radicn(j(k))minus1 biki
It follows from Lemma 31 that Ek
is dominated by En(j(k))
radicn(j(k))minus1 aij(k)i
for any k and so Lemma 45 im-
plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof
References
[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998
[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature
bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J
Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J
Differential Geom 54 (2000) no 1 37ndash74
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
HIGH-DIMENSIONAL ELLIPSOIDS 27
[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace
operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of
Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x
[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures
on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121
[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-
ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047
[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates
[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]
[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-
ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146
[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001
[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur
les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed
[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal
transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903
[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)
[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz
order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-
sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282
[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-
tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580
[19] R Ozawa and T Shioya Limit formulas for metric measure invariants
and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2
[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-
ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0
[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]
[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-
28 DAISUKE KAZUKAWA AND TAKASHI SHIOYA
infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]
[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8
[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7
[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures
[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287
[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel
and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y
[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower
curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x
Mathematical Institute Tohoku University Sendai 980-8578 JAPAN
E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp
- 1 Introduction
- 2 Preliminaries
-
- 21 Distance between measures
- 22 mm-Isomorphism and Lipschitz order
- 23 Observable diameter
- 24 Box distance and observable distance
- 25 Pyramid
- 26 Dissipation
- 27 Asymptotic concentration
- 28 Gaussian space
-
- 3 Weak convergence of ellipsoids
- 4 Box convergence of ellipsoids
- References
-