Probability Theory and Related Fields manuscript No.(will be inserted by the editor)

Faces of Poisson–Voronoi Mosaics

Daniel Hug · Rolf Schneider

Received: date / Accepted: date

Abstract For a stationary Poisson–Voronoi tessellation in Euclidean d-space and for

k ∈ 1, . . . ,d, we consider the typical k-dimensional face with respect to a natural

centre function. We express the distribution of this typical k-face in terms of a certain

Poisson process of closed halfspaces in a k-dimensional space. Then we show that,

under the condition of large inradius, the relative boundary of the typical k-face lies,

with high probability, in a narrow spherical annulus.

Keywords Poisson–Voronoi tessellation · typical k-face · spherical shape

Mathematics Subject Classification (2000) 60D05 · 52A20

1 Introduction

The Voronoi tessellation induced by a stationary Poisson point process in Euclidean

space Rd is probably the most intensively studied special model of a random mosaic.

In low dimensions, it has many applications in various areas of science (see the hints

in the books by Stoyan, Kendall and Mecke [15] and by Okabe, Boots, Sugihara and

Daniel Hug

Karlsruhe Institute of Technology

Department of Mathematics

Kaiserstr. 89–93

D-76128 Karlsruhe

Germany

E-mail: daniel.hug@kit.edu

Rolf Schneider

Mathematisches Institut

Albert-Ludwigs-Universitat

Eckerstr.1

D-79104 Freiburg i. Br.

Germany

E-mail: rolf.schneider@math.uni-freiburg.de

2

Chiu [11]), and in general it offers quite a number of mathematical challenges. We

refer to the lecture notes by Møller [10] and the references in Section 10.3 of [14].

In [1], Baumstark and Last have obtained fairly general distributional properties

of stationary Poisson–Voronoi tessellations. The present paper continues such inves-

tigations, but is somehow complementary to [1]. A k-dimensional face F in a Voronoi

tessellation is (in general) the intersection of d − k+ 1 cells of the tessellation, and

the nuclei of these cells are called the neighbours of F . These neighbours lie on the

relative boundary of a (d − k)-dimensional ball. We call the radius of this ball the

co-radius of F and its centre the generalized nucleus of F . This point lies in the affine

hull of F , but not necessarily in F (except if k = d or k = 0). For the weighted typical

k-face of a stationary Poisson–Voronoi tessellation and, to a lesser extent, for the typ-

ical k-face, Baumstark and Last have unified and extended several earlier results and

have established various distributional properties of the neighbours. Our focus here

is on the distribution of the typical k-face itself. We find representations (as explicitly

as possible) for the joint distribution of the typical k-face and its co-radius (Theo-

rem 2), as well as for the regular conditional distribution of the typical k-face, given

the co-radius. This is expressed in terms of the intersection of a Poisson process of

k-dimensional closed halfspaces, with explicitly given intensity measure (Theorem

3).

We use this information to obtain a result on the approximate spherical shape of

typical k-faces with large inradius in a stationary Poisson–Voronoi tessellation. Re-

sults of this type were first obtained by Calka [2], who determined the joint distribu-

tion of inradius and circumradius of the typical cell of a two-dimensional stationary

Poisson–Voronoi mosaic and deduced that, under the condition that the inradius is

equal to r, the boundary of the typical cell is, with high probability, contained in a

circular ring of width tending to zero with r tending to infinity (the approach does not

extend to higher dimensions). Further related limit theorems in the plane were proved

by Calka und Schreiber [3]. In [4], the same authors remark that similar results for

the typical Poisson–Voronoi cell in higher dimensions can be deduced from their ob-

tained results on random polytopes in balls and will be developed in future work. It

appears that lower-dimensional typical faces require other methods. With a different

approach (adapted from the work [5], [6] on extensions of D.G. Kendall’s problem),

we establish here a similar result on typical k-faces.

Let Z(k) denote the typical k-face (where k ∈ 1, . . . ,d) of a stationary Poisson–

Voronoi tessellation in Rd , with respect to the generalized nucleus as centre function

(precise definitions are given in Section 2). We denote by Bd the unit ball with centre

at the origin and by affK the affine hull of a set K in Rd . For a convex body K we

define

Rm(K) := maxr ≥ 0 : rBd ∩ affK ⊂ Kif o ∈ K, otherwise we formally set Rm(K) =−1, and we put

RM(K) := minr ≥ 0 : K ⊂ rBd.

Then we define

R(k)m := Rm(Z

(k)), R(k)M := RM(Z(k)).

3

Theorem 1 Let ρ ≥ 1, let k ∈ 1, . . . ,d, and choose β with

0 < β <2d− k− 1

k+ 1, so that α := d− (1+β )

k+ 1

2> 0.

Then there exist constants c1, depending only on the dimension d and on a positive

lower bound for the intensity γ of the underlying Poisson process, and c2, depending

only on the dimension d, such that

P

R(k)M > ρ +ρ−β | R

(k)m ≥ ρ

≤ c1 exp [−c2γρα ] .

In the assumption ρ ≥ 1 one could choose a different positive lower bound for

ρ ; this would affect only the constants. Note that the thickness ρ−β of the annulus

containing, with high probability, the relative boundary of Z(k) decreases if the di-

mension k becomes smaller. In the special case k = d = 2 considered in [2], we have

0 < β < 1/3 and α = 12(1− 3β ), which is consistent with [2, Th. 5].

A few explanations concerning Theorem 1 are in order. The theorem expresses,

in a precise way, the following information about the shape of the typical k-face Z(k)

with respect to the generalized nucleus as centre function. Under the condition that

Z(k) contains the ball ρB(k) := ρ(Bd ∩ affZ(k)), it even satisfies

B(k) ⊂ 1

ρZ(k) ⊂

(1+

1

ρ1+β

)B(k)

with overwhelming probability; thus Z(k) is approximately a ball, with increasing

precision as ρ →∞. Theorem 1 belongs, therefore, to a group of results on asymptotic

shapes of cells and faces in random mosaics under the condition of large size, which

was initiated by a problem of D.G. Kendall (for this, see [15, p. ix]). Kendall’s original

problem concerned the zero cell of an isotropic stationary Poisson line tessellation in

the plane, under the condition of large area. Later extensions measured the size also

by other functionals, and extended the problem in various ways, also to Poisson–

Voronoi tessellations. For these, in [5] the asymptotic shape of large typical cells

(with respect to the nucleus) was determined, with large size either measured by

the volume (or an intrinsic volume) or the inradius with respect to the nucleus. In

contrast to the case of stationary Poisson hyperplane tessellations, where results on

asymptotic shapes of (weighted or unweighted) typical k-faces under the condition

of large k-volume were proved in [7] for k < d, it remains an open problem whether

similar results hold for Poisson–Voronoi mosaics, with the size of typical k-faces

measured by k-volume. If, however, size is measured by the inradius with respect to

the generalized nucleus, then Theorem 1 provides such a result, giving generalized

variants of the case of cells treated in [5] and of the estimates in the planar case

obtained by Calka [2]. That the generalized nucleus plays a particular role in the

geometry of faces in Poisson–Voronoi mosaics was already clear, for example from

the distributional results of Baumstark and Last [1]. It is, therefore, not surprising

that the assumption that the generalized nucleus of a face is deep inside the face,

has strong consequences for the geometry of the face. Of course, the assumption

R(k)m ≥ 0 implies that the typical k-face contains its generalized nucleus. We compute

4

the probability of this event in a remark after Theorem 3. It should be observed that

the condition of large inradius with respect to the generalized nucleus is not only

a condition of large size, as in the Kendall type problems, but also a condition on

the position of the face with respect to its generalized nucleus. Whether for k < d

there are asymptotic shapes of typical k-faces in Poisson–Voronoi mosaics under size

conditions alone, remains open.

We give a brief outline of the following sections. First we recall the construction

of the faces of a Voronoi mosaic and define the co-radius of a face as the distance of

the affine hull of the face from its neighbouring nuclei. Using Palm distributions, we

define the typical k-face and the typical k-co-radius of the Voronoi mosaic induced

by a stationary Poisson process X . Theorem 2 expresses the joint distribution of these

random objects by an integral over R×G(d,d − k), involving a random polytope

Z(L,r; X)⊂ L⊥ which is defined for each subspace L in the Grassmannian G(d,d−k)and each positive number r. In Section 3 this random polytope is then identified as

the intersection of a Poisson process of closed halfspaces, and the main purpose of

that section is to determine the intensity measure of this process (Theorem 3). Alto-

gether, the obtained information on the distribution of the typical k-face is sufficient

to represent the conditional distribution appearing in Theorem 1 in a way allowing

the required estimation. This is done in Section 4, where a few more explanations

will be given.

2 The distribution of the typical k-face

We work in d-dimensional Euclidean space Rd (d ≥ 2), with scalar product 〈·, ·〉 and

induced norm ‖ · ‖. Unit ball and unit sphere are denoted, respectively, by Bd and

Sd−1. Lebesgue measure on Rd is denoted by λ and spherical Lebesgue measure on

Sd−1 by σ . The space G(d,k) is the Grassmannian of k-dimensional linear subspaces,

and νk is its rotation invariant probability measure. For L ∈ G(d,k), let SL := Sd−1∩L

denote the unit sphere in L, λL the k-dimensional Lebesgue measure on L, and σL the

(k−1)-dimensional spherical Lebesgue measure on SL. We write λ1 for the Lebesgue

measure on R. The space of convex bodies (nonempty, compact, convex subsets) of

Rd is denoted by K ; it is equipped with the Hausdorff metric. For a topological space

T we denote by B(T ) its Borel σ -algebra. Measures, without further specification,

are Borel measures.

We need first a description of the k-faces of a Voronoi mosaic and recall some

facts from [14, p. 472]. Let η be a subset of Rd which is admissible. By this we

mean that η is locally finite, in general position (that is, any p+1 points of η are not

contained in a (p−1)-dimensional plane, p = 1, . . . ,d), in general quadratic position

(that is, no d + 2 points of η lie on some sphere), and the convex hull of η is Rd . A

realization of a stationary Poisson point process in Rd with positive intensity is a.s.

admissible. By V (η) we denote the Voronoi mosaic generated by η (that is, the set

of Voronoi cells of η). Let k ∈ 0, . . . ,d, and let x0, . . . ,xd−k ∈ η be d − k+ 1 dis-

tinct points. There is a unique closed ball Bd−k(x0, . . . ,xd−k) of dimension d − k that

contains these points in its relative boundary. We denote by z(x0, . . . ,xd−k) its centre

and by E(x0, . . . ,xd−k) the k-dimensional affine subspace through z(x0, . . . ,xd−k) and

5

orthogonal to the affine hull of Bd−k(x0, . . . ,xd−k). Let B0(y,r) denote the open ball

with centre y and radius r and

S(x0, . . . ,xd−k;η) := y ∈ E(x0, . . . ,xd−k) : B0(y,‖y− x0‖)∩η = /0.

If S(x0, . . . ,xd−k;η) 6= /0, then F = S(x0, . . . ,xd−k;η) is a k-face of V (η), and every

k-face of V (η) is obtained in this way, for uniquely determined points x0, . . . ,xd−k.

We put z(x0, . . . ,xd−k) =: z(F,η) and call this point the generalized nucleus of F

(though it need not be contained in F); for k = d it is the nucleus of F . The radius of

the ball Bd−k(x0, . . . ,xd−k) is called the co-radius of F and denoted by R(F,η).If V (η) contains a unique k-face F with z(F,η) = o, we denote this face by

Fko (η).

Now let X be a stationary Poisson point process in Rd with intensity γ > 0. Let

X (d) =V (X) be the Voronoi tessellation generated by X , let X (k) be the process of its

k-faces and γ(k) the intensity of X (k), for k = 0, . . . ,d (the explicit values of γ(k) for

d = 2,3 are listed in [14, Th. 10.2.5]; for d = 4, see the remark at the top of page 68

in [9]). In particular, γ(d) = γ; cf. [14, (10.32)].

We denote the underlying probability space by (Ω ,A,P) (and suppose in the fol-

lowing that it is large enough), and E denotes mathematical expectation. By

Nk := ∑F∈X(k)

δz(F,X),

where δ denotes a Dirac measure, we define a stationary random measure on Rd and

denote by P0Nk

the Palm distribution of X with respect to Nk. Let Y be a point pro-

cess (defined on the underlying probability space) with distribution P0Nk

. The Voronoi

mosaic V (Y ) has a.s. a unique k-face with generalized nucleus at o. We define

Z(k) := Fko (Y ), R(k) := R(Z(k),Y )

and call Z(k) the typical k-face and R(k) the typical k-co-radius of X (d), both with

respect to the generalized nucleus as centre function. (Note that the terminology ‘of

X (d)’ is correct, since X (d) determines X .)

For the joint distribution of Z(k) and R(k) we obtain (e.g., by the line of reasoning

leading to [12, formula (6)]), for A ∈ B(K ), I ∈ B([0,∞)) and B ∈ B(Rd) with

λ (B) = 1,

P

Z(k) ∈ A, R(k) ∈ I

= P

0Nk

(η : Fk

o (η) ∈ A, R(Fko (η),η) ∈ I

)

=1

γ(k)E ∑

F∈X(k)

1A(F − z(F, X))1I(R(F, X))1B(z(F, X)). (1)

In formulating (1), we have used that the intensity γNkof the stationary random

measure Nk is equal to γ(k). This is not immediately obvious, since z(F, X) depends

not only on the face F but also on the process X , and it may have arbitrarily large dis-

tance from F . We prove the equality here by adapting an argument from [13, p. 138].

Let c denote the centre function defined by the centre of the smallest circumscribed

6

ball (as used in [14, pp. 100 ff]). Let Cd0 := [0,1)d be the ‘half-open unit cube’, and

let (zi)i∈N be an enumeration of Zd . Then, using the stationarity of X ,

γNk= E ∑

F∈X(k)

1z(F, X) ∈Cd0

= ∑i∈N

E ∑F∈X(k)

1c(F) ∈Cd0 + zi1z(F, X) ∈Cd

0

= ∑i∈N

E ∑G+zi∈X(k)

1c(G) ∈Cd01z(G+ zi, X) ∈Cd

0

= ∑i∈N

E ∑G∈(X−zi)(k)

1c(G) ∈Cd01z(G, X − zi) ∈Cd

0 − zi

= ∑i∈N

E ∑G∈X(k)

1c(G) ∈Cd01z(G, X) ∈Cd

0 − zi

= E ∑G∈X(k)

1c(G) ∈Cd0= γ(k).

Since the early work of Miles, the use of integral geometric transformations in

stochastic geometry is well established. For the convenience of the reader, we first

state such a transformation formula which is needed below. Its application in this

context goes back to Miles [8] and was repeated in Møller [9] and [14, Sect. 10.2]. If

k ∈ 1, . . . ,d − 1 and f is a nonnegative measurable function on (Rd)d−k+1, then

∫

(Rd)d−k+1f dλ d−k+1 = bd(d−k)((d − k)!)k+1

∫

G(d,d−k)

∫

Rd

∫ ∞

0

∫

SL

. . .∫

SL

f (z+ ru0, . . . ,z+ rud−k)∆k+1d−k (u0, . . . ,ud−k)r

d(d−k)−1

× σL(du0) · · ·σL(dud−k)dr λ (dz)νd−k(dL). (2)

Here ∆d−k(u0, . . . ,ud−k) is the (d − k)-dimensional volume of the convex hull of

u0, . . . ,ud−k. For the value of the constant bd(d−k) we refer to [14, (7.8)]. The transfor-

mation formula is obtained by combining the affine Blaschke–Petkantschin formula

([14, Theorem 7.2.7]) and [14, Theorem 7.3.1], a transformation formula involving

spheres.

We turn to the joint distribution of Z(k) and R(k), for k ∈ 1, . . . ,d − 1, and use

(1) to obtain, for A ∈ B(K ), I ∈ B([0,∞)) and a Borel set B ⊂ Rd with λ (B) = 1,

γ(k)P

Z(k) ∈ A, R(k) ∈ I

= E ∑F∈X(k)

1A(F − z(F, X))1I(R(F, X))1B(z(F, X))

7

=1

(d − k+ 1)!E ∑

(x0,...,xd−k)∈Xd−k+16=

1A

(S(x0, . . . ,xd−k; X)− z(x0, . . . ,xd−k)

)

× 1I

(R(S(x0, . . . ,xd−k; X), X)

)1B(z(x0, . . . ,xd−k)),

where Xm6= denotes the set of ordered m-tuples of pairwise distinct points from X .

(Note that the elements of A are nonempty by definition, so that 1A( /0) = 0.) We use

the Slivnyak–Mecke formula ([14, Cor. 3.2.3]) and then the transformation formula

(2). Since X has intensity measure γλ , this gives

γ(k)P

Z(k) ∈ A, R(k) ∈ I

=γd−k+1

(d− k+ 1)!E

∫

(Rd)d−k+1

1A

(S(x0, . . . ,xd−k; X ∪x0, . . . ,xd−k)− z(x0, . . . ,xd−k)

)

× 1I

(R(S(x0, . . . ,xd−k; X ∪x0, . . . ,xd−k), X ∪x0, . . . ,xd−k)

)

× 1B(z(x0, . . . ,xd−k))λd−k+1(d(x0, . . . ,xd−k))

=bd(d−k)((d − k)!)kγd−k+1

d− k+ 1

∫

G(d,d−k)

∫

Rd

∫ ∞

0

∫

SL

. . .∫

SL

E1A

(S

(z+ ru0, . . . ,z+ rud−k; X ∪

d−k⋃

i=0

z+ rui)− z

)1I(r)1B(z)

× ∆ k+1d−k(u0, . . . ,ud−k)r

d(d−k)−1 σL(du0) · · ·σL(dud−k)dr λ (dz)νd−k(dL).

Now we observe that

S

(z+ ru0, . . . ,z+ rud−k; X ∪

d−k⋃

i=0

z+ rui)− z = S

(ru0, . . . ,rud−k; X − z

)

and that S(ru0, . . . ,rud−k;η), for r > 0 and points u0, . . . ,ud−k ∈ L in general position,

depends only on L,r and η ; we denote it, therefore, by Z(L,r;η). Thus,

Z(L,r;η) =

y ∈ L⊥ : B0

(y,√‖y‖2 + r2

)∩η = /0

(3)

8

for any admissible set η . Since X is stationary, we obtain

γ(k)P

Z(k) ∈ A, R(k) ∈ I

=bd(d−k)((d − k)!)kγd−k+1

d− k+ 1

∫

G(d,d−k)

∫

I

∫

Sd−k+1L

E1A(Z(L,r; X))

× ∆ k+1d−k(u0, . . . ,ud−k)r

d(d−k)−1 σd−k+1L (d(u0, . . . ,ud−k))dr νd−k(dL)

=bd(d−k)((d − k)!)kγd−k+1

d− k+ 1S(d− k,d− k,k+ 1)

×∫

I

∫

G(d,d−k)E1A(Z(L,r; X ))νd−k(dL)rd(d−k)−1 dr,

where [14, Theorem 8.2.3] was used and the number S(d−k,d−k,k+1) is given by

[14, (8.5)]. This constant as well as bd(d−k) can be expressed in terms of κd := λ (Bd)

and ωd := σ(Sd−1) = dκd (also for other dimensions). We state the obtained result

as a theorem.

Theorem 2 For k ∈ 1, . . . ,d−1, the joint distribution of the typical k-face Z(k) and

the typical k-co-radius R(k) of the Poisson–Voronoi tessellation induced by X is given

by

P

Z(k) ∈ A, R(k) ∈ I

=C(d,k)γd−k+1

γ(k)

∫

I

∫

G(d,d−k)P

Z(L,r; X ) ∈ A

νd−k(dL)rd(d−k)−1 dr

for A ∈ B(K ) and I ∈ B([0,∞)), where Z(L,r;η) is defined by (3) and

C(d,k) =1

(d − k+ 1)!ωd−k

d+1ωk+1

κ(d−k)d−2

κ(d−k+1)(d−1).

In particular, for the distribution of the typical k-co-radius we obtain the repre-

sentation

PR(k)(I) = P

R(k) ∈ I

=C(d,k)

γd−k+1

γ(k)

∫

IP

Z(L,r; X) 6= /0

rd(d−k)−1 dr.

Moreover,

P

Z(L,r; X) 6= /0

≥ P

B0(o,r)∩ X = /0

> 0,

and therefore we can also write

P

Z(k) ∈ A, R(k) ∈ I

=

∫

I

∫

G(d,d−k)P

Z(L,r; X ) ∈ A | Z(L,r; X) 6= /0

νd−k(dL)P

R(k)(dr).

9

Hence, the regular conditional distribution of the typical k-face under the hypothesis

R(k) = r is given by

P

Z(k) ∈ A | R(k) = r

=

∫

G(d,d−k)P

Z(L,r; X) ∈ A | Z(L,r; X) 6= /0

νd−k(dL)

for λ1 almost all r > 0. If the Borel set A is invariant under rotations, then

P

Z(k) ∈ A | R(k) = r

= P

Z(L,r; X) ∈ A | Z(L,r; X) 6= /0

,

where L ∈ G(d,d−k) is arbitrary and fixed. A similar remark applies to the statement

of Theorem 2.

3 Induced Poisson processes in subspaces

Theorem 2 expresses, in particular, the distribution of the typical k-face, for k ∈1, . . . ,d − 1, as a mixture of the distributions PZ(L,r; X) ∈ ·. In the present sec-

tion we investigate, therefore, the random set Z(L,r; X ).Let L∈G(d,d−k) and r > 0 be given, for k ∈ 1, . . . ,d−1. As before, η denotes

an admissible point set. For y ∈ L⊥ we have

y ∈ Z(L,r;η) ⇔∀x ∈ η : x /∈ B0

(y,√‖y‖2 + r2

),

and for x ∈ η ,

x ∈ B0

(y,√‖y‖2 + r2

)⇔ ‖x− y‖2 < ‖y‖2 + r2 ⇔ ‖x‖2 − 2〈x,y〉< r2.

We decompose x = x1 + x2 with x1 ∈ L⊥ and x2 ∈ L. For x /∈ L we put

ux :=x1

‖x1‖and τx :=

‖x‖2 − r2

2‖x1‖.

For u ∈ SL⊥ and t ∈ R we define

H(u, t) := w ∈ L⊥ : 〈w,u〉= t, H−(u, t) := w ∈ L⊥ : 〈w,u〉 ≤ t. (4)

With this notation, for x /∈ L we have

x /∈ B0

(y,√

‖y‖2 + r2

)⇔ 〈y,ux〉 ≤ τx ⇔ y ∈ H−(ux,τx).

The stationary Poisson process X satisfies P

X ∩L 6= /0= 0, hence

Z(L,r; X) =⋂

x∈X\L

H−(ux,τx) a.s.

10

Therefore, we define a point process XL,r in SL⊥ ×R by

XL,r := (ux,τx) : x ∈ X \L.

Then Z(L,r; X ) is given by

Z(L,r; X) =⋂

(u,t)∈XL,r

H−(u, t) =: ZL,r. (5)

b

b

b

bb

b

b

b

L

x0

r

r

x1

o

x

x1

y

τxux

SL⊥

ux

L⊥

Explanation: The sphere with centre y ∈ L⊥ through the points x0, x1 ∈ L, which are at distance r

from o, contains the point x. The shaded halfspace w ∈ L⊥ : 〈ux,w〉 ≤ τx is the set Z(L,r;x).

The subsequent Theorem 3 shows, in particular, that the intensity measure

EXL,r =: ΘL,r of the process XL,r is locally finite and does not have atoms, hence

XL,r is a.s. simple and thus, by its definition, a Poisson process.

Theorem 3 For k ∈ 1, . . . ,d−1, the intensity measure of the Poisson point process

XL,r is given by

ΘL,r(A) = γωd−krd

∫

SL⊥

∫ ∞

−∞

∫ π/2

01A

(u,

rτ

cosω

)P(τ)

× cosk−1 ω sind−k−1 ω dω dτ σL⊥(du)

for A ∈ B(SL⊥ ×R), where

P(τ) :=

(τ +

√1+ τ2

)d

√1+ τ2

.

11

Proof Let A ∈B(SL⊥ ×R). Let G : Rd \L → SL⊥ ×R be defined by G(x) := (ux,τx).Then

ΘL,r(A) = γ

∫

Rd1A(G(x))λ (dx)

= γ

∫

L⊥

∫

L1A(G(x1 + x2))λL(dx2)λL⊥(dx1).

We introduce polar coordinates in L\ o and in L⊥ \ o, writing

x1 = tu, x2 = sv, t,s > 0, u ∈ SL⊥ , v ∈ SL,

and thus

G(tu+ sv) =

(u,

t2 + s2 − r2

2t

).

Substituting rt for t and rs for s, we get

ΘL,r(A) = γωd−krd

∫

SL⊥

∫ ∞

0

∫ ∞

01A

(u,r

t2 + s2 − 1

2t

)tk−1sd−k−1 dsdt σL⊥(du).

We introduce new coordinates (τ,ω) ∈ R× (0,π/2) by means of the transformation

T : R× (0,π/2)→ (0,∞)2, (τ,ω) 7→ λ (τ,ω)(cosω ,sinω),

where

λ (τ,ω) := τ cosω +√

1+ τ2 cos2 ω .

For fixed ω ∈ (0,π/2), the function τ 7→ λ (τ,ω) is strictly increasing, and we have

limτ→−∞

λ (τ,ω) = 0, limτ→∞

λ (τ,ω) = ∞.

This clearly implies that T is bijective. For the Jacobian JT of T we get

JT (τ,ω) = |det(DT (τ,ω))|= λτ(τ,ω)λ (τ,ω)

with

λτ(τ,ω) :=∂λ

∂τ(τ,ω) =

cosω√1+ τ2 cos2 ω

λ (τ,ω),

hence JT (τ,ω)> 0. Moreover, with (t,s) = T (τ,ω) we have (t2 + s2 −1)/(2t) = τ .

Therefore, the transformation formula applied with the diffeomorphism T yields

∫ ∞

0

∫ ∞

01A

(u,r

t2 + s2 − 1

2t

)tk−1sd−k−1 dsdt

=

∫ ∞

−∞

∫ π/2

01A(u,rτ) [λ (τ,ω)cosω ]k−1 [λ (τ,ω)sinω ]d−k−1

×λτ(τ,ω)λ (τ,ω)dω dτ

=

∫ ∞

−∞

∫ π/2

01A(u,rτ)

λ (τ,ω)d

√1+ τ2 cos2 ω

cosk ω sind−k−1 ω dω dτ.

12

Using Fubini’s theorem twice and substituting τ/cosω for τ , we get

∫ ∞

0

∫ ∞

01A

(u,r

t2 + s2 − 1

2t

)tk−1sd−k−1 dsdt

=

∫ π/2

0

∫ ∞

−∞1A(u,rτ)

(τ cosω +

√1+ τ2 cos2 ω

)d

√1+ τ2 cos2 ω

cosk ω sind−k−1 ω dτ dω

=

∫ π/2

0

∫ ∞

−∞1A

(u,

rτ

cosω

)(

τ +√

1+ τ2)d

√1+ τ2

cosk−1 ω sind−k−1 ω dτ dω

=

∫ ∞

−∞

∫ π/2

01A

(u,

rτ

cosω

)(

τ +√

1+ τ2)d

√1+ τ2

cosk−1 ω sind−k−1 ω dω dτ,

which yields the required equation. ⊓⊔

Remark 1 Let σ0L⊥ := ω−1

k σL⊥ and define the function pr : R→ [0,∞) by

pr(τ) := γωkωd−krd−1∫ π/2

0P(cosω

rτ)

cosk ω sind−k−1 ω dω .

Then the measure µr with

µr(B) :=∫

Bpr(τ)dτ

for B ∈ B(R) satisfies ΘL,r = σ0L⊥ ⊗ µr. Therefore, XL,r is an independently marked

Poisson point process in R with mark space SL⊥ , for which the intensity measure of

the underlying unmarked point process is µr and the mark distribution is σ0L⊥ . This

fact will not be used in the following.

Of the function P appearing in Theorem 3 we shall need the indefinite integral.

We have ∫ x

−∞P(t)dt =

1

dF(x)

with

F(x) =(

x+√

1+ x2)d

,

since F ′(x) = dP(x) and limx→−∞ F(x) = 0. The function F is strictly increasing and

strictly convex.

Remark 2 It is now easy to see that the random set ZL,r defined by (5) is a.s.

bounded, for example, by modifying the proof of [14, Theorem 10.3.2] and using

that∫ ∞−∞ P(t)dt = ∞. From

∫ a−∞ P(t)dt < ∞ for a ∈ R it then follows that ZL,r, if not

empty, is a.s. a polytope.

13

From Theorem 3, we draw a conclusion for the intensity measure ΘL,r evaluated

at special sets. We start with the case k ∈ 1, . . . ,d − 1. The set of convex bodies in

L⊥ is denoted by KL⊥ . For K ∈ KL⊥ let

HK := (u, t) ∈ SL⊥ ×R : K 6⊂ H−(u, t) (6)

and define

ΦL,r(K) :=ΘL,r(HK).

Then, denoting by h(K, ·) the support function of K, we obtain from Theorem 3

ΦL,r(K) = γωd−krd

∫

SL⊥

∫ ∞

−∞

∫ π/2

01HK

(u,

rt

cosω

)P(t)

× cosk−1 ω sind−k−1 ω dω dt σL⊥(du)

= γωd−krd

∫

SL⊥

∫ π/2

0

∫ ∞

−∞1 rt

cosω< h(K,u)

P(t)dt

× cosk−1 ω sind−k−1 ω dω σL⊥(du).

We have ∫ ∞

−∞1 rt

cosω< h(K,u)

P(t)dt =

1

dF

(cosω

rh(K,u)

)

and hence

ΦL,r(K) =γωd−k

drd

∫ π/2

0

[∫

SL⊥

F

(cosω

rh(K,u)

)σL⊥(du)

](7)

× cosk−1 ω sind−k−1 ω dω .

This will be needed in the next section.

Remark 3 The obtained results allow us to compute the probability that the typical k-

face contains its generalized nucleus. From Theorem 2 we get, with a fixed subspace

L ∈ G(d,d− k),

P

o ∈ Z(k)

=C(d,k)

γd−k+1

γ(k)

∫ ∞

0P

o ∈ Z(L,r; X)

rd(d−k)−1 dr,

and here

P

o ∈ Z(L,r; X)

=P

XL,r(Ho) = 0

= exp

[−ΘL,r(Ho)

]= exp [−ΦL,r(o)] .

Together with (7) and F(0) = 1 this leads to

P

o ∈ Z(k)

=

1

(d − k+ 1)(d− k)d

(ωd+1

κd

)d−k

ωk+1

κ(d−k)d−2

κ(d−k+1)(d−1)

γ

γ(k).

For example, for d = 2 we have Po ∈ Z(1) = 2/3, and for d = 3 we obtain that

Po ∈ Z(1)= 105/192≈ 0.5469 and Po ∈ Z(2)= 140/(35+ 24π2)≈ 0.5151.

14

We add a few remarks concerning the case k= d. Here we have L= o, L⊥ =Rd ,

and

Z(o,0; X) := y ∈ Rd : B0(y,‖y‖)∩ X = /0

P-a.s. is the Voronoi cell of X ∪o with nucleus o. Then, again P-a.s., we have

Z(o,0; X) =⋂

x∈X\oH−(

x

‖x‖ ,‖x‖2

)=: Zo,0 = Z(d).

Here and in the following, definitions and results in the case k = d often correspond

to definitions and results in the cases where k ∈ 1, . . . ,d − 1 if we choose r = 0.

The simple point process

Xo,0 :=

(x

‖x‖ ,‖x‖2

): x ∈ X \ o

is a Poisson point process in Sd−1 × (0,∞) ⊂ Sd−1 × R with intensity measure

Θo,0 := EXo,0. For a convex body K ∈ K with o ∈ K we define

HK := (u, t) ∈ Sd−1 × (0,∞) : K 6⊂ H−(u, t).

Then, as in [5, Section 2], we get

Φo,0(K) :=Θo,0(HK) = 2dγU(K)

with

U(K) :=1

d

∫

Sd−1h(K,u)d σ(du). (8)

4 Typical faces with large inradius

The purpose of this section is to prove Theorem 1. The conditional probability appear-

ing in that theorem can now, thanks to the representation of the distribution of Z(k)

that we have obtained, be expressed in a form allowing further conclusions. Let ρ ≥ 1

and 0 < ε ≤ 1 be given. Further, let k ∈ 1, . . . ,d. We fix a subspace L ∈ G(d,d−k)and denote by

BL⊥ := Bd ∩L⊥

its unit ball. By Theorem 2 and rotation invariance we have

P

R(k)m ≥ ρ

= P

Rm(Z

(k))≥ ρ

= C(d,k)γd−k+1

γ(k)

∫ ∞

0PρBL⊥ ⊂ ZL,rrd(d−k)−1 dr

= C(d,k)γd−k+1

γ(k)

∫ ∞

0exp [−ΦL,r(ρBL⊥)] r

d(d−k)−1 dr.

15

Similarly,

P

R(k)m ≥ ρ , R

(k)M > ρ + s

= P

Rm(Z

(k))≥ ρ , RM(Z(k))> ρ + s

=C(d,k)γd−k+1

γ(k)

∫ ∞

0PρBL⊥ ⊂ ZL,r 6⊂ (ρ + s)BL⊥rd(d−k)−1 dr

and hence

P

R(k)M > ρ + s | R

(k)m ≥ ρ

=

∫ ∞0 PρBL⊥ ⊂ ZL,r 6⊂ (ρ + s)BL⊥rd(d−k)−1 dr

∫ ∞0 exp [−ΦL,r(ρBL⊥)]r

d(d−k)−1 dr. (9)

To estimate (9) from above, we first fix the number r, with r > 0 if k ≤ d−1, and

r = 0 if k = d (in this case the integration is omitted). We work in the k-dimensional

subspace L⊥ as basic space. Recall that H−(u, t), defined by (4), is a closed halfspace

in L⊥ and that

ZL,r =⋂

(u,t)∈XL,r

H−(u, t)

is a random closed convex set in L⊥ (possibly empty).

Put ‖K‖ := max‖x‖ : x ∈ K for K ∈KL⊥ . For ρ ≥ 1, s > 0 and m ∈N we define

Kρ ,s(m) := K ∈ KL⊥ : ρBL⊥ ⊂ K 6⊂ (ρ + s)BL⊥ , ‖K‖ ∈ ρ(m,m+ 1]

and

qρ ,s(m) := PZL,r ∈ Kρ ,s(m).Then we have

PρBL⊥ ⊂ ZL,r 6⊂ (ρ + s)BL⊥= ∑m∈N

qρ ,s(m). (10)

The reason for decomposing in this way (and thus for introducing the parameter m)

lies in the fact that it allows us to exploit a fundamental property of Poisson processes

([14, Th. 3.2.2(b)]), permitting a reduction to a finite number N of independent, iden-

tically distributed objects, in the following way.

Let

Cρ ,m := ρ(m+ 1)BL⊥.

If N ≤ k, then P

ZL,r ∈ Kρ ,s(m), XL,r(HCρ,m) = N

= 0. Hence, we have (using the

notation (6))

qρ ,s(m) =∞

∑N=k+1

PXL,r(HCρ,m ) = NpN

with

pN := PZL,r ∈ Kρ ,s(m) | XL,r(HCρ,m ) = N

16

and

PXL,r(HCρ,m ) = N=ΘL,r(HCρ,m )

N

N!exp[−ΘL,r(HCρ,m)

],

since XL,r is a Poisson process. For (u1, t1), . . . ,(uN , tN) ∈ SL⊥ ×R we use the abbre-

viation

P((u, t)(N)) :=N⋂

i=1

H−(ui, ti).

Then we have, by the Poisson property ([14, Th. 3.2.2(b)]) mentioned above,

pN =1

ΦL,r(Cρ ,m)N

∫

H NCρ,m

1P((u, t)(N)) ∈ Kρ ,s(m)Θ NL,r(d((u1, t1), . . . ,(uN , tN))).

We need an upper estimate for (10), hence for qρ ,s(m) and, therefore, for pN .

For sufficiently large m, this estimate is provided in Lemma 1 and based on the fact

that P((u, t)(N)) ∈ Kρ ,s(m) implies for m large that the intersection P((u, t)(N)) con-

tains a long segment with one endpoint at o, which happens with small probabil-

ity. In the course of the proof of Lemma 1, we first obtain an estimate in terms of

ΦL,r(Kρ ,m), for certain convex bodies Kρ ,m. The estimation of ΦL,r(Kρ ,m) then makes

use of the distributional results obtained in the previous sections, in particular of (7).

Since qρ ,s(m)≤ qρ ,0(m), we need only the case s = 0 in Lemma 1. The estimation for

small numbers m, in Lemma 3, requires an estimate for the increase of the functional

ΦL,r when passing from a ball with centre o to the convex hull of this ball and a point

outside; this is provided by Lemma 2.

In this paper, c1,c2, . . . denote constants depending only on the dimension d and

possibly on a lower bound γ0 > 0 for the intensity γ . We use repeatedly that a constant

depending on k ∈ 1, . . . ,d can be estimated by a constant depending only on d.

Lemma 1 There exists a number m0, depending only on d, such that

qρ ,0(m)≤ c3 exp [−ΦL,r(ρBL⊥)]exp[−c4γmdρd

]

for m ≥ m0 and ρ ≥ 1.

Proof Suppose that (u1, t1), . . . ,(uN , tN) ∈ HCρ,m are such that P := P((u, t)(N)) ∈Kρ ,0(m). Then P has a vertex v with ‖v‖ ≥ ρm. This vertex is the intersection of k

facets of P. Hence, there exists an index set J ⊂ 1, . . . ,N with k elements such that

v=⋂

j∈J

H(u j, t j).

We denote the segment [o,v] by S = S((u j, t j), j ∈ J) and its length by |S|. The seg-

ment S satisfies

S ⊂ H−(ui, ti) for i ∈ 1, . . . ,N.Since S ⊂Cρ ,m, we have

∫

HCρ,m

1S ⊂ H−(u, t)ΘL,r(d(u, t)) = ΦL,r(Cρ ,m)−ΦL,r(S)

≤ ΦL,r(Cρ ,m)−ΦL,r(Sρm),

17

where we have denoted by Sρm a fixed segment in L⊥ with endpoints o and ρme,

‖e‖= 1, and we have used that ΦL,r is invariant under rotations of L⊥ in itself and is

increasing under set inclusion. We obtain

pN ≤(

N

k

)ΦL,r(Cρ ,m)

−N

∫

H kCρ,m

1∣∣S((u j, t j), j ∈ 1, . . . ,k)

∣∣ ≥ ρm

×∫

HN−k

Cρ,m

1

S((u j, t j), j ∈ 1, . . . ,k)⊂ H−(ui, ti) for i = k+ 1, . . . ,N

×Θ N−kL,r (d((uk+1, tk+1), . . . ,(uN , tN)))Θ k

L,r(d((u1, t1), . . . ,(uk, tk)))

≤(

N

k

)ΦL,r(Cρ ,m)

−N

∫

H kCρ,m

[ΦL,r(Cρ ,m)−ΦL,r(Sρm)]N−k

×Θ kL,r(d((u1, t1), . . . ,(uk, tk)))

=

(N

k

)ΦL,r(Cρ ,m)

k−N[ΦL,r(Cρ ,m)−ΦL,r(Sρm)

]N−k.

This yields

qρ ,0(m) ≤∞

∑N=k+1

[ΦL,r(Cρ ,m)]N

N!exp[−ΦL,r(Cρ ,m)

]

×(

N

k

)ΦL,r(Cρ ,m)

k−N[ΦL,r(Cρ ,m)−ΦL,r(Sρm)

]N−k

=1

k![ΦL,r(Cρ ,m)]

k exp[−ΦL,r(Cρ ,m)

]

×∞

∑N=k+1

1

(N − k)!

[ΦL,r(Cρ ,m)−ΦL,r(Sρm)

]N−k

≤ 1

k![ΦL,r(Cρ ,m)]

k exp[−ΦL,r(Sρm)

]. (11)

Let k ∈ 1, . . . ,d− 1. Since F is convex, Jensen’s inequality leads from (7) to

ΦL,r(Sρm) =γωd−k

drd

∫ π/2

0

[∫

SL⊥

F(cosω

rρm〈e,u〉+

)σL⊥(du)

]

× cosk−1 ω sind−k−1 ω dω

≥ γωd−kωk

drd

∫ π/2

0F

(1

ωk

∫

SL⊥

cosω

rρm〈e,u〉+σL⊥(du)

)

× cosk−1 ω sind−k−1 ω dω ,

18

hence

ΦL,r(Sρm)≥γωd−kωk

drd

∫ π/2

0F(cosω

rρmc(k)

)cosk−1 ω sind−k−1 ω dω

with a constant c(k) depending only on k. For x ≥ 0 and a ≥ 1 we have

rdF(

ax

r

)=

(ax+

√r2 +(ax)2

)d

=d

∑j=0

(d

j

)(ax)d− j(r2 +(ax)2) j/2

≥ (ad − 1)xd + xd + d[(ad−1 − 1)xd−1 + xd−1](r2 + x2)12

+d

∑j=2

(d

j

)xd− j(r2 + x2) j/2

≥ rdF(x

r

)+(ad − 1)xd + d(ad−1− 1)xd−1r.

Inserting this with x = ρ cosω and a= mc(k), we conclude that there exists a number

m0, depending only on d, such that for m ≥ m0 we have

ΦL,r(Sρm)≥ ΦL,r(ρBL⊥)+ γ[c5mdρd + c6md−1ρd−1r

].

This inequality is also true, by a simpler direct estimate based on the functional U

from (8), if k = d and r = 0. Together with (11) this gives

qρ ,0(m) ≤ 1

k!

[ΦL,r(Cρ ,m)

]kexp [−ΦL,r(ρBL⊥)]

×exp[−c6γmd−1ρd−1r

]exp[−c5γmdρd

]

for k ∈ 1, . . . ,d. We need an upper estimate for ΦL,r(Cρ ,m). Again we first consider

the case k ∈ 1, . . . ,d − 1. Using F(x)≤ (1+ 2x)d for x ≥ 0, we get

ΦL,r(Cρ ,m) = ΦL,r(ρ(m+ 1)BL⊥)

=γωd−kωk

drd

∫ π/2

0F(cosω

rρ(m+ 1)

)cosk−1 ω sind−k−1 ω dω

≤ γωd−kωk

d

∫ π/2

0(r+ 2ρ(m+ 1)cosω)d cosk−1 ω sind−k−1 ω dω

=γωd−kωk

d

d

∑j=0

(d

j

)2 j(m+ 1) jρ jrd− j

Γ(

j+k2

)Γ(

d−k2

)

Γ(

d+ j2

)

and hence

ΦL,r(Cρ ,m)≤ c7γd

∑j=0

m jρ jrd− j. (12)

19

This estimate is also true for k = d and r = 0, again by a simple direct argument.

From this, we obtain

[ΦL,r(Cρ ,m)]k exp

[−c6γmd−1ρd−1r

]exp[−(c5/2)γmdρd

]

≤[

c7γd

∑j=0

m jρ jrd− j

]k

exp[−c6γmd−1ρd−1r

]exp[−(c5/2)γmdρd

]

≤((

c7γmdρd)k

exp[−(c5/2)γmdρd

])(

d

∑j=0

rd− j

)k

exp [−c6γ0r]

,

since m ≥ 1, ρ ≥ 1, γ ≥ γ0 > 0 and r > 0 if k ∈ 1, . . . ,d − 1 (respectively, r = 0 if

k = d). The right-hand side attains a maximum, which depends only on d and a lower

bound γ0 for γ . This yields the assertion of the lemma. ⊓⊔

Remark 4 Similar arguments as above show that PR(k)m ≥ ρ→ 0 exponentially fast

as ρ → ∞. In fact, from

rdF(ρ

rcosω

)≥ rd +ρd cosd ω

we get

ΦL,r(ρBL⊥)≥ γ(a1rd + a2ρd)

with constants a1,a2 depending only on d. Hence we deduce that

P

R(k)m ≥ ρ

≤ a3

γ

γ(k)exp(−γa2ρd

),

with a constant a3 depending only on d.

For x ∈ L⊥ \ρBL⊥ we have ΦL,r(conv(ρBL⊥ ∪x))>ΦL,r(ρBL⊥), by the mono-

tonicity of ΦL,r. For our estimation of qρ ,s(m) for m ≤ m0 in Lemma 3 we need

a careful improvement of this inequality in terms of ‖x‖− ρ . This is provided by

Lemma 2.

Lemma 2 Let ρ ≥ 1 and 0 ≤ ε ≤ 1. If x ∈ L⊥ satisfies ‖x‖ ≥ (1+ ε)ρ , then

ΦL,r(conv(ρBL⊥ ∪x))≥ ΦL,r(ρBL⊥)+ γg(ρ ,r)εk+1

2

with

g(ρ ,r) = maxc8ρd ,c9ρrd−1,

where c8,c9 are suitable constants depending only on d.

20

Proof We may assume that x = (1+ ε)ρe with e ∈ L⊥, ‖e‖= 1 and ε > 0.

Let k ∈ 1, . . . ,d − 1. By (7),

Ψ(ε) := ΦL,r(conv(ρBL⊥ ∪x))−ΦL,r(ρBL⊥)

=γωd−k

drd

∫ π/2

0

∫

SL⊥

[F(cosω

rh(conv(ρBL⊥ ∪x),u)

)−F

(cosω

rρ)]

× σL⊥(du)cosk−1 ω sind−k−1 ω dω .

First, we consider the cases where k ∈ 2, . . . ,d − 1. Decomposing u ∈ SL⊥ in the

form u = te+√

1− t2 u0 with u0 ∈ SL⊥ ∩ e⊥, we get

Ψ (ε) =γωd−kωk−1

drd

∫ π/2

0

∫ 1

11+ε

[F

(cosω

r(1+ ε)ρt

)−F

(cosω

rρ)]

× (1− t2)k−3

2 dt cosk−1 ω sind−k−1 ω dω .

We write

f (ε) :=

∫ 1

11+ε

[F

(cosω

r(1+ ε)ρt

)−F

(cosω

rρ)]

(1− t2)k−3

2 dt.

Observing that the integrand vanishes at the lower boundary of the integration inter-

val, we obtain

f ′(ε) = d

∫ 1

11+ε

P(cosω

r(1+ ε)ρt

) cosω

rρt(1− t2)

k−32 dt. (13)

Next we use the estimate P(x)> (2x)d−1 for x ≥ 0. This gives

f ′(ε) ≥ d2d−1(cosω

rρ)d∫ 1

11+ε

td(1− t2)k−3

2 dt.

The substitution 1− t2 = τ leads to

∫ 1

11+ε

td(1− t2)k−3

2 dt =1

2

∫ ε(2+ε)

(1+ε)2

0τ

k−32 (1− τ)

d−12 dτ.

For 0 ≤ y ≤ 1, a > 0 and b ≥ 1 we can estimate∫ y

0ta−1(1− t)b−1dt ≥ c(a,b)ya.

In fact, for 0 ≤ y ≤ 1/2 we have 1− t ≥ 1/2 in the integrand and, therefore,

∫ y

0ta−1(1− t)b−1dt ≥ 1

2b−1

∫ y

0ta−1dt =

1

a2b−1ya.

For 1/2 ≤ y ≤ 1 trivially

∫ y

0ta−1(1− t)b−1dt ≥

∫ 1/2

0ta−1(1− t)b−1dt = c′(a,b) ·1 ≥ c′(a,b)ya.

21

Therefore, we obtain

∫ 1

11+ε

td(1− t2)k−3

2 dt ≥ c10

(ε(2+ ε)

(1+ ε)2

) k−12

≥ c10

(3

4ε

) k−12

≥ c11εk−1

2 .

We conclude that

Ψ ′(ε)≥ c12γρdεk−1

2 = g′(ε)

with g(ε) := c12γρd 2k+1

εk+1

2 . From (Ψ −g)′(ε)≥ 0 for 0≤ ε ≤ 1 and (Ψ −g)(0) = 0

we obtain Ψ − g ≥ 0, which gives

Ψ (ε)≥ c13γρdεk+1

2 .

We start again from (13), but now we use the estimate P(x)≥ 1 for x ≥ 0. In the

same way as before, this leads to the estimate

Ψ(ε) ≥ c14γρrd−1εk+1

2 .

Both estimates together yield the assertion if k ∈ 2, . . . ,d− 1.

If k = 1, then

Ψ(ε) =γωd−1

drd

∫ π/2

0

[F(cosω

r(1+ ε)ρ

)−F

(cosω

rρ)]

sind−2 ω dω

and hence

Ψ ′(ε) = γωd−1rd

∫ π/2

0P

(cosω

r(1+ ε)ρ

) cosω

rρ sind−2 ω dω .

Now the proof can be completed by estimating P as before.

Finally, the same conclusion is obtained for k = d with a simplified argument. ⊓⊔

Lemma 3 If s = ερ with 0 ≤ ε ≤ 1, then

qρ ,s(m)≤ 1

k![ΦL,r(Cρ ,m)]

k exp[−ΦL,r(ρBL⊥)− γg(ρ ,r)ε

k+12

].

Proof Suppose that (u1, t1), . . . ,(uN , tN) ∈ HCρ,m are such that P := P((u, t)(N)) ∈Kρ ,s(m). Then P has a vertex x with ‖x‖ ≥ ρ + s = ρ(1+ ε). This vertex is the

intersection of k facets of P. Hence, there exists an index set J ⊂ 1, . . . ,N with k

elements such that

x=⋂

j∈J

H(u j, t j).

By Lemma 2,

∫

HCρ,m

1conv(ρBL⊥ ∪x)⊂ H−(u, t)ΘL,r(d(u, t))

≤ ΦL,r(Cρ ,m)−ΦL,r(ρBL⊥)− γg(ρ ,r)εk+1

2 .

22

We write x = x((u1, t1), . . . ,(uk, tk)) and obtain

PZL,r ∈ Kρ ,s(m) | XL,r(HCρ,m) = N

≤(

N

k

)ΦL,r(Cρ ,m)

−N

∫

H kCρ,m

1‖x((u1, t1), . . . ,(uk, tk))‖ ≥ ρ(1+ ε)

∫

HN−k

Cρ,m

1conv(ρBL⊥ ∪x((u1, t1), . . . ,(uk, tk)))⊂ H(ui, ti), i = k+ 1, . . . ,N

×Θ N−kL,r (d((uk+1, tk+1), . . . ,(uN , tN)))Θ k

L,r(d((u1, t1), . . . ,(uk, tk)))

≤(

N

k

)ΦL,r(Cρ ,m)

k−N[ΦL,r(Cρ ,m)−ΦL,r(ρBL⊥)− γg(ρ ,r)ε

k+12

]N−k

.

As before, summation over N gives the assertion. ⊓⊔

Proof of Theorem 1. We have to estimate the conditional probability given by (9).

Using Lemma 3 for m ≤ m0 and Lemma 1 for m > m0 and putting s = ερ , we get

PρBL⊥ ⊂ ZL,r 6⊂ (ρ + s)BL⊥= ∑m∈N

qρ ,s(m)≤ A(ρ ,r,ε)+B(ρ ,r)

with

A(ρ ,r,ε) :=m0

∑m=1

1

k![ΦL,r(Cρ ,m)]

k exp[−ΦL,r(ρBL⊥)− γg(ρ ,r)ε

k+12

]

and

B(ρ ,r) := ∑m>m0

c3 exp [−ΦL,r(ρBL⊥)]exp[−c4γmdρd

].

The sum

∑m>m0

exp[−c4γ(md − 1)ρd

]

is convergent and bounded by a constant depending only on d and γ0 (the lower bound

for γ introduced before Lemma 1), since ρ ≥ 1. We conclude that

B(ρ ,r)≤ c15 exp [−ΦL,r(ρBL⊥)]exp[−c4γρd

].

Before the estimation of A(ρ ,r,ε), we specify ε , choosing

ε = ρ−(1+β ), s = ερ = ρ−β , 0 < β <2d− k− 1

k+ 1;

then

α := d− (1+β )k+ 1

2> 0

and, by the definition of g(ρ ,r),

g(ρ ,r)εk+1

2 ≥ c8ρdεk+1

2 = c8ρα

23

and

g(ρ ,r)εk+1

2 ≥ c9ρrd−1εk+1

2 = c9ρα

(r

ρ

)d−1

.

From (12) we get

m0

∑m=1

1

k!

[ΦL,r(Cρ ,m)

]kexp

[−1

2γg(ρ ,r)ε

k+12

]

≤m0

∑m=1

1

k!

[c7γ

d

∑j=0

m jρ jrd− j

]k

exp

[−1

2γg(ρ ,r)ε

k+12

]

≤ c16

[γ

d

∑j=0

ρd

(r

ρ

)d− j]k

exp

[−1

2γρα max

c8,c9

(r

ρ

)d−1]

≤ c16

((γρα)k exp

[−c8

6γρα

])(ρ (d−α)k exp

[−c8

6γ0ρα

])

×(

d

∑j=0

(r

ρ

)d− j)k

exp

[−c9

6γ0

(r

ρ

)d−1]≤ c17.

The rearrangements in the second to last estimate show that all bounds depend only

on d and on the lower bound γ0 for γ .

We conclude that

A(ρ ,r,ε)≤ c17 exp [−ΦL,r(ρBL⊥)]exp

[−1

2c8γρα

].

It follows that

PρBL⊥ ⊂ ZL,r 6⊂ (ρ + s)BL⊥

≤ exp [−ΦL,r(ρBL⊥)]

(c17 exp

[−1

2c8γρα

]+ c15 exp

[−c4γρd

])

≤ exp [−ΦL,r(ρBL⊥)]c18 exp [−c19γρα ] . (14)

From (9) we now deduce that

P

R(k)M > ρ +ρ−β | R

(k)m ≥ ρ

≤ c18 exp [−c19γρα ] .

For k = d we have

PR(d)m ≥ ρ= exp

[−Φo,0(ρBd)

]

and (14) remains true with L = o and r = 0, by a simpler argument. This concludes

the proof in all cases. ⊓⊔

Acknowledgements We thank the referee for a suggestion that led to a simplification of the proof of

Theorem 3.

24

References

1. Baumstark, V., Last, G.: Some distributional results for Poisson–Voronoi tessellations. Adv. in Appl.

Probab. (SGSA) 39, 16–40 (2007)

2. Calka, P.: The distributions of the smallest disks containing the Poisson–Voronoi typical cell and the

Crofton cell in the plane. Adv. in Appl. Probab. (SGSA) 34, 702–717 (2002)

3. Calka, P., Schreiber, T.: Limit theorems for the typical Poisson–Voronoi cell and the Crofton cell with

a large inradius. Ann. Probab. 33, 1625–1642 (2005)

4. Calka, P., Schreiber, T.: Large deviation probabilities for the number of vertices of random polytopes

in the ball. Adv. in Appl. Probab. (SGSA) 38, 47–58 (2006)

5. Hug, D., Reitzner, M., Schneider, R.: Large Poisson–Voronoi cells and Crofton cells. Adv. in Appl.

Probab. (SGSA) 36, 667–690 (2004)

6. Hug, D., Schneider, R.: Asymptotic shapes of large cells in random tessellations. Geom. Funct. Anal.

17, 156–191 (2007)

7. Hug, D., Schneider, R.: Large faces in Poisson hyperplane mosaics. Ann. Probab. (to appear)

8. Miles, R.E.: Sectional Voronoi tessellations. Rev. Un. Mat. Argentina 29, 310–327 (1984)

9. Møller, J.: Random tessellations in Rd . Adv. in Appl. Probab. 21, 37–73 (1989)

10. Møller, J.: Lectures on Random Voronoi Tessellations. Lecture Notes in Statist. 87, Springer, New

York (1994)

11. Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations; Concepts and Applications of

Voronoi Diagrams. 2nd ed., Wiley, Chichester (2000)

12. Schneider, R.: Weighted faces of Poisson hyperplane tessellations. Adv. in Appl. Probab. (SGSA) 41,

682–694 (2009)

13. Schneider, R., Weil, W.: Stochastische Geometrie. Teubner, Stuttgart, Leipzig (2000)

14. Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin, Heidelberg (2008)

15. Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications. 2nd ed., Wiley,

Chichester (1995)