SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig...
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SL(n) invariant notions of surface area
Monika Ludwig
Polytechnic Institute of New York University
AMS Sectional Meeting, Wesleyan University, 2008
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Felix Klein’s Erlangen Program 1872
Geometry is the study of invariants of transformation groups.
Groups acting on Rn
• Group of rigid motions: x 7→ Ux + bwhere U is an orthogonal n × n matrix and b ∈ Rn
• Special linear group SL(n): x 7→ Axwhere A is an n × n matrix of determinant 1
• General linear group GL(n): x 7→ Axwhere A is an n × n matrix of determinant 6= 0
• Special affine group SA(n): x 7→ Ax + bwhere A is an n × n matrix of determinant 1 and b ∈ Rn
![Page 3: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/3.jpg)
Felix Klein’s Erlangen Program 1872
Geometry is the study of invariants of transformation groups.
Groups acting on Rn
• Group of rigid motions: x 7→ Ux + bwhere U is an orthogonal n × n matrix and b ∈ Rn
• Special linear group SL(n): x 7→ Axwhere A is an n × n matrix of determinant 1
• General linear group GL(n): x 7→ Axwhere A is an n × n matrix of determinant 6= 0
• Special affine group SA(n): x 7→ Ax + bwhere A is an n × n matrix of determinant 1 and b ∈ Rn
![Page 4: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/4.jpg)
Felix Klein’s Erlangen Program 1872
Geometry is the study of invariants of transformation groups.
Groups acting on Rn
• Group of rigid motions: x 7→ Ux + bwhere U is an orthogonal n × n matrix and b ∈ Rn
• Special linear group SL(n): x 7→ Axwhere A is an n × n matrix of determinant 1
• General linear group GL(n): x 7→ Axwhere A is an n × n matrix of determinant 6= 0
• Special affine group SA(n): x 7→ Ax + bwhere A is an n × n matrix of determinant 1 and b ∈ Rn
![Page 5: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/5.jpg)
Felix Klein’s Erlangen Program 1872
Geometry is the study of invariants of transformation groups.
Groups acting on Rn
• Group of rigid motions: x 7→ Ux + bwhere U is an orthogonal n × n matrix and b ∈ Rn
• Special linear group SL(n): x 7→ Axwhere A is an n × n matrix of determinant 1
• General linear group GL(n): x 7→ Axwhere A is an n × n matrix of determinant 6= 0
• Special affine group SA(n): x 7→ Ax + bwhere A is an n × n matrix of determinant 1 and b ∈ Rn
![Page 6: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/6.jpg)
Felix Klein’s Erlangen Program 1872
Geometry is the study of invariants of transformation groups.
Groups acting on Rn
• Group of rigid motions: x 7→ Ux + bwhere U is an orthogonal n × n matrix and b ∈ Rn
• Special linear group SL(n): x 7→ Axwhere A is an n × n matrix of determinant 1
• General linear group GL(n): x 7→ Axwhere A is an n × n matrix of determinant 6= 0
• Special affine group SA(n): x 7→ Ax + bwhere A is an n × n matrix of determinant 1 and b ∈ Rn
![Page 7: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/7.jpg)
Felix Klein’s Erlangen Program 1872
Geometry is the study of invariants of transformation groups.
Groups acting on Rn
• Group of rigid motions: x 7→ Ux + bwhere U is an orthogonal n × n matrix and b ∈ Rn
• Special linear group SL(n): x 7→ Axwhere A is an n × n matrix of determinant 1
• General linear group GL(n): x 7→ Axwhere A is an n × n matrix of determinant 6= 0
• Special affine group SA(n): x 7→ Ax + bwhere A is an n × n matrix of determinant 1 and b ∈ Rn
![Page 8: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/8.jpg)
Valuations
• Kn space of convex bodies (compact convex sets) in Rn
• A functional Φ : Kn → R is a valuation ⇐⇒
Φ(K ) + Φ(L) = Φ(K ∪ L) + Φ(K ∩ L)
for all K , L ∈ Kn such that K ∪ L ∈ Kn.
• Hilbert’s Third Problem:Dehn 1902, Sydler 1965, Jessen & Thorup 1978, Sah 1979,McMullen 1989, . . .
• Theory of valuations:Blaschke 1937, Hadwiger 1949, Sallee 1966, Schneider 1971,Groemer 1972, McMullen 1977, Goodey & Weil 1984, Betke& Kneser 1985, Klain 1995, Ludwig 1999, Reitzner 1999,Alesker 1999, Fu 2006, Bernig 2006, Haberl 2006, Schuster2006, . . .
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Valuations
• Kn space of convex bodies (compact convex sets) in Rn
• A functional Φ : Kn → R is a valuation ⇐⇒
Φ(K ) + Φ(L) = Φ(K ∪ L) + Φ(K ∩ L)
for all K , L ∈ Kn such that K ∪ L ∈ Kn.
• Hilbert’s Third Problem:Dehn 1902, Sydler 1965, Jessen & Thorup 1978, Sah 1979,McMullen 1989, . . .
• Theory of valuations:
Blaschke 1937, Hadwiger 1949, Sallee 1966, Schneider 1971,Groemer 1972, McMullen 1977, Goodey & Weil 1984, Betke& Kneser 1985, Klain 1995, Ludwig 1999, Reitzner 1999,Alesker 1999, Fu 2006, Bernig 2006, Haberl 2006, Schuster2006, . . .
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Valuations
• Kn space of convex bodies (compact convex sets) in Rn
• A functional Φ : Kn → R is a valuation ⇐⇒
Φ(K ) + Φ(L) = Φ(K ∪ L) + Φ(K ∩ L)
for all K , L ∈ Kn such that K ∪ L ∈ Kn.
• Hilbert’s Third Problem:Dehn 1902, Sydler 1965, Jessen & Thorup 1978, Sah 1979,McMullen 1989, . . .
• Theory of valuations:
Blaschke 1937, Hadwiger 1949, Sallee 1966, Schneider 1971,Groemer 1972, McMullen 1977, Goodey & Weil 1984, Betke& Kneser 1985, Klain 1995, Ludwig 1999, Reitzner 1999,Alesker 1999, Fu 2006, Bernig 2006, Haberl 2006, Schuster2006, . . .
![Page 11: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/11.jpg)
Valuations
• Kn space of convex bodies (compact convex sets) in Rn
• A functional Φ : Kn → R is a valuation ⇐⇒
Φ(K ) + Φ(L) = Φ(K ∪ L) + Φ(K ∩ L)
for all K , L ∈ Kn such that K ∪ L ∈ Kn.
• Hilbert’s Third Problem:Dehn 1902, Sydler 1965, Jessen & Thorup 1978, Sah 1979,McMullen 1989, . . .
• Theory of valuations:
Blaschke 1937, Hadwiger 1949, Sallee 1966, Schneider 1971,Groemer 1972, McMullen 1977, Goodey & Weil 1984, Betke& Kneser 1985, Klain 1995, Ludwig 1999, Reitzner 1999,Alesker 1999, Fu 2006, Bernig 2006, Haberl 2006, Schuster2006, . . .
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Hadwiger’s Classification Theorem 1952
TheoremA functional Φ : Kn → R is a continuous, rigid motion invariantvaluation
⇐⇒
∃ c0, c1, . . . , cn ∈ R such that
Φ(K ) = c0 V0(K ) + · · ·+ cn Vn(K )
for every K ∈ Kn.
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Hadwiger’s Classification Theorem 1952
TheoremA functional Φ : Kn → R is a continuous, rigid motion invariantvaluation
⇐⇒
∃ c0, c1, . . . , cn ∈ R such that
Φ(K ) = c0 V0(K ) + · · ·+ cn Vn(K )
for every K ∈ Kn.
V0(K ), . . . ,Vn(K ) intrinsic volumes of K
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Hadwiger’s Classification Theorem 1952
TheoremA functional Φ : Kn → R is a continuous, rigid motion invariantvaluation
⇐⇒
∃ c0, c1, . . . , cn ∈ R such that
Φ(K ) = c0 V0(K ) + · · ·+ cn Vn(K )
for every K ∈ Kn.
V0(K ), . . . ,Vn(K ) intrinsic volumes of K
Vn n-dimensional volume2Vn−1 surface area
V0 Euler characteristic (V0(K ) = 1 for K 6= ∅, V0(∅) = 0)
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Hadwiger’s Classification Theorem 1952
TheoremA functional Φ : Kn → R is a continuous, rigid motion invariantvaluation
⇐⇒
∃ c0, c1, . . . , cn ∈ R such that
Φ(K ) = c0 V0(K ) + · · ·+ cn Vn(K )
for every K ∈ Kn.
V0(K ), . . . ,Vn(K ) intrinsic volumes of K
New proof by Daniel Klain 1995
“Introduction to Geometric Probability” by Klain and Rota 1997
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Equi-affine surface area of K ∈ Kn
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx
κ(K , x) Gaussian curvature at x ∈ ∂K
Definition for smooth surfacesGeorg Pick 1914“Vorlesungen uber Differentialgeometrie II”by Wilhelm Blaschke 1923
Definition for general convex bodies
Kurt Leichtweiß 1986, Carsten Schutt &Elisabeth Werner 1990, Erwin Lutwak 1991,Georg Dolzmann & Daniel Hug 1995
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Equi-affine surface area of K ∈ Kn
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx
κ(K , x) Gaussian curvature at x ∈ ∂K
Definition for smooth surfacesGeorg Pick 1914“Vorlesungen uber Differentialgeometrie II”by Wilhelm Blaschke 1923
Definition for general convex bodies
Kurt Leichtweiß 1986, Carsten Schutt &Elisabeth Werner 1990, Erwin Lutwak 1991,Georg Dolzmann & Daniel Hug 1995
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Equi-affine surface area of K ∈ Kn
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx
κ(K , x) Gaussian curvature at x ∈ ∂K
Definition for smooth surfacesGeorg Pick 1914“Vorlesungen uber Differentialgeometrie II”by Wilhelm Blaschke 1923
Definition for general convex bodies
Kurt Leichtweiß 1986, Carsten Schutt &Elisabeth Werner 1990, Erwin Lutwak 1991,Georg Dolzmann & Daniel Hug 1995
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SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
0
• uK (x) outer unit normalvector of K at x
• dµK (x) = x · uK (x) dxcone measure on ∂K
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SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
0
uK (x)
• uK (x) outer unit normalvector of K at x
• dµK (x) = x · uK (x) dxcone measure on ∂K
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SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
0
x · uK (x)
uK (x)
• uK (x) outer unit normalvector of K at x
• dµK (x) = x · uK (x) dxcone measure on ∂K
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SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
0
x · uK (x)
uK (x)
• uK (x) outer unit normalvector of K at x
• dµK (x) = x · uK (x) dxcone measure on ∂K
![Page 23: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/23.jpg)
SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
x · uK (x)
a = x · uK (x)
br
r = b2
a = κ(K , x)−1
n−1
V (EK (x)) = Vn(Bn) a bn−1
• uK (x) outer unit normalvector of K at x
• κ0(K , x) =κ(K , x)
(x · uK (x))n+1
• EK (x) osculating centeredellipsoid at x
• κ0(K , x) =Vn(Bn)
Vn(EK (x))2
• Gheorghe Titeica 1908
![Page 24: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/24.jpg)
SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
x · uK (x)
EK (x)
a = x · uK (x)
br
r = b2
a = κ(K , x)−1
n−1
V (EK (x)) = Vn(Bn) a bn−1
• uK (x) outer unit normalvector of K at x
• κ0(K , x) =κ(K , x)
(x · uK (x))n+1
• EK (x) osculating centeredellipsoid at x
• κ0(K , x) =Vn(Bn)
Vn(EK (x))2
• Gheorghe Titeica 1908
![Page 25: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/25.jpg)
SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
x · uK (x)
EK (x)
x
a = x · uK (x)
br
r = b2
a = κ(K , x)−1
n−1
V (EK (x)) = Vn(Bn) a bn−1
• uK (x) outer unit normalvector of K at x
• κ0(K , x) =κ(K , x)
(x · uK (x))n+1
• EK (x) osculating centeredellipsoid at x
• κ0(K , x) =Vn(Bn)
Vn(EK (x))2
• Gheorghe Titeica 1908
![Page 26: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/26.jpg)
SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
EK (x)
x
a = x · uK (x)
b
r
r = b2
a = κ(K , x)−1
n−1
V (EK (x)) = Vn(Bn) a bn−1
• uK (x) outer unit normalvector of K at x
• κ0(K , x) =κ(K , x)
(x · uK (x))n+1
• EK (x) osculating centeredellipsoid at x
• κ0(K , x) =Vn(Bn)
Vn(EK (x))2
• Gheorghe Titeica 1908
![Page 27: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/27.jpg)
SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
x
a = x · uK (x)
br
r = b2
a = κ(K , x)−1
n−1
V (EK (x)) = Vn(Bn) a bn−1
• uK (x) outer unit normalvector of K at x
• κ0(K , x) =κ(K , x)
(x · uK (x))n+1
• EK (x) osculating centeredellipsoid at x
• κ0(K , x) =Vn(Bn)
Vn(EK (x))2
• Gheorghe Titeica 1908
![Page 28: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/28.jpg)
SL(n) invariance of Ω
Ω(K ) =
∫∂K
κ(K , x)1
n+1 dx =
∫∂K
κ0(K , x)1
n+1 dµK (x)
x
K
x
a = x · uK (x)
br
r = b2
a = κ(K , x)−1
n−1
V (EK (x)) = Vn(Bn) a bn−1
• uK (x) outer unit normalvector of K at x
• κ0(K , x) =κ(K , x)
(x · uK (x))n+1
• EK (x) osculating centeredellipsoid at x
• κ0(K , x) =Vn(Bn)
Vn(EK (x))2
• Gheorghe Titeica 1908
![Page 29: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/29.jpg)
Properties of Ω : Kn → R
Ω(K ) =
∫∂K
κ0(K , x)1
n+1 dµK (x)
• Ω is equi-affine invariant (SA(n) invariant)
Ω(AK + b) = Ω(K ) ∀A ∈ SL(n), b ∈ Rn
• Ω is upper semicontinuous (Lutwak 1991)
Ω(K ) ≥ lim supj→∞
Ω(Kj)
if Kj → K as j →∞.
• Ω is a valuation (Schutt 1993)
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Properties of Ω : Kn → R
Ω(K ) =
∫∂K
κ0(K , x)1
n+1 dµK (x)
• Ω is equi-affine invariant (SA(n) invariant)
Ω(AK + b) = Ω(K ) ∀A ∈ SL(n), b ∈ Rn
• Ω is upper semicontinuous (Lutwak 1991)
Ω(K ) ≥ lim supj→∞
Ω(Kj)
if Kj → K as j →∞.
• Ω is a valuation (Schutt 1993)
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Properties of Ω : Kn → R
Ω(K ) =
∫∂K
κ0(K , x)1
n+1 dµK (x)
• Ω is equi-affine invariant (SA(n) invariant)
Ω(AK + b) = Ω(K ) ∀A ∈ SL(n), b ∈ Rn
• Ω is upper semicontinuous (Lutwak 1991)
Ω(K ) ≥ lim supj→∞
Ω(Kj)
if Kj → K as j →∞.
• Ω is a valuation (Schutt 1993)
![Page 32: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/32.jpg)
Theorem (Ludwig 1999, Ludwig & Reitzner 1999)
A functional Φ : Kn → R is an upper semicontinuous, equi-affineinvariant valuation
⇐⇒
∃ c0, c1 ∈ R, c2 ≥ 0 such that
Φ(K ) = c0 V0(K ) + c1 Vn(K ) + c2 Ω(K )
for every K ∈ Kn.
![Page 33: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/33.jpg)
Applications and recent results
• PDEAndrews 1996, Angenent & Sapiro & Tannenbaum 1998,Chen & Howard & Lutwak & Yang & Zhang 2004, Trudinger& Wang 2000, Sheng & Trudinger & Wang 2006, Wang 2002
• Pattern recognition and image processingAngenent 1998, Calabi & Olver & Tannenbaum 1996,Dionisio & Kim 2004, Flusser & Suk 1993, Moisan 1998,Olver 1997, Sapiro & Tannenbaum 1993, Sapiro 1993
• Stochastic geometryBarany 1997, Barany & Prodromou 2006, Boroczky &Reitzner 2004, Reitzner 2002, Schutt & Werner 2000
• Discrete and convex geometryBarany 1997, Cheung & Zhao 2008, Grinberg & Zhang 1999,Petrov 2006, Schmuckenschlager 1995, Stancu 2006, Vershik& Zeitouni 1999
• Polytopal approximation
![Page 34: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/34.jpg)
Applications and recent results
• PDEAndrews 1996, Angenent & Sapiro & Tannenbaum 1998,Chen & Howard & Lutwak & Yang & Zhang 2004, Trudinger& Wang 2000, Sheng & Trudinger & Wang 2006, Wang 2002
• Pattern recognition and image processingAngenent 1998, Calabi & Olver & Tannenbaum 1996,Dionisio & Kim 2004, Flusser & Suk 1993, Moisan 1998,Olver 1997, Sapiro & Tannenbaum 1993, Sapiro 1993
• Stochastic geometryBarany 1997, Barany & Prodromou 2006, Boroczky &Reitzner 2004, Reitzner 2002, Schutt & Werner 2000
• Discrete and convex geometryBarany 1997, Cheung & Zhao 2008, Grinberg & Zhang 1999,Petrov 2006, Schmuckenschlager 1995, Stancu 2006, Vershik& Zeitouni 1999
• Polytopal approximation
![Page 35: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/35.jpg)
Applications and recent results
• PDEAndrews 1996, Angenent & Sapiro & Tannenbaum 1998,Chen & Howard & Lutwak & Yang & Zhang 2004, Trudinger& Wang 2000, Sheng & Trudinger & Wang 2006, Wang 2002
• Pattern recognition and image processingAngenent 1998, Calabi & Olver & Tannenbaum 1996,Dionisio & Kim 2004, Flusser & Suk 1993, Moisan 1998,Olver 1997, Sapiro & Tannenbaum 1993, Sapiro 1993
• Stochastic geometryBarany 1997, Barany & Prodromou 2006, Boroczky &Reitzner 2004, Reitzner 2002, Schutt & Werner 2000
• Discrete and convex geometryBarany 1997, Cheung & Zhao 2008, Grinberg & Zhang 1999,Petrov 2006, Schmuckenschlager 1995, Stancu 2006, Vershik& Zeitouni 1999
• Polytopal approximation
![Page 36: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/36.jpg)
Applications and recent results
• PDEAndrews 1996, Angenent & Sapiro & Tannenbaum 1998,Chen & Howard & Lutwak & Yang & Zhang 2004, Trudinger& Wang 2000, Sheng & Trudinger & Wang 2006, Wang 2002
• Pattern recognition and image processingAngenent 1998, Calabi & Olver & Tannenbaum 1996,Dionisio & Kim 2004, Flusser & Suk 1993, Moisan 1998,Olver 1997, Sapiro & Tannenbaum 1993, Sapiro 1993
• Stochastic geometryBarany 1997, Barany & Prodromou 2006, Boroczky &Reitzner 2004, Reitzner 2002, Schutt & Werner 2000
• Discrete and convex geometryBarany 1997, Cheung & Zhao 2008, Grinberg & Zhang 1999,Petrov 2006, Schmuckenschlager 1995, Stancu 2006, Vershik& Zeitouni 1999
• Polytopal approximation
![Page 37: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/37.jpg)
Applications and recent results
• PDEAndrews 1996, Angenent & Sapiro & Tannenbaum 1998,Chen & Howard & Lutwak & Yang & Zhang 2004, Trudinger& Wang 2000, Sheng & Trudinger & Wang 2006, Wang 2002
• Pattern recognition and image processingAngenent 1998, Calabi & Olver & Tannenbaum 1996,Dionisio & Kim 2004, Flusser & Suk 1993, Moisan 1998,Olver 1997, Sapiro & Tannenbaum 1993, Sapiro 1993
• Stochastic geometryBarany 1997, Barany & Prodromou 2006, Boroczky &Reitzner 2004, Reitzner 2002, Schutt & Werner 2000
• Discrete and convex geometryBarany 1997, Cheung & Zhao 2008, Grinberg & Zhang 1999,Petrov 2006, Schmuckenschlager 1995, Stancu 2006, Vershik& Zeitouni 1999
• Polytopal approximation
![Page 38: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/38.jpg)
Polytopal Approximation
• Kn convex bodies in Rn
• Pm convex polytopes with m vertices
• K4P = (K ∪ P)\(K ∩ P)symmetric difference of K and P
• δ(K ,Pm) = minVn(K4P) : P ∈ Pm
Fejes Toth 1948, McClure & Vitale 1975, Schneider 1986, Gruber1988, 1993 (P ⊂ K ), Ludwig 1999:∃ cn > 0
limm→∞
δ(K ,Pm)m2
n−1 = cn Ω(K )n+1n−1
for K ∈ Kn, ∂K ∈ C2
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Polytopal Approximation
• Kn convex bodies in Rn
• Pm convex polytopes with m vertices
• K4P = (K ∪ P)\(K ∩ P)symmetric difference of K and P
• δ(K ,Pm) = minVn(K4P) : P ∈ Pm
Fejes Toth 1948, McClure & Vitale 1975, Schneider 1986, Gruber1988, 1993 (P ⊂ K ), Ludwig 1999:∃ cn > 0
limm→∞
δ(K ,Pm)m2
n−1 = cn Ω(K )n+1n−1
for K ∈ Kn, ∂K ∈ C2
![Page 40: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/40.jpg)
Polytopal Approximation
• Kn0 convex bodies in Rn, 0 ∈ intK
• Banach-Mazur distanceδBM(K ,P) =minλ > 1 : P ⊂ AK ⊂ λP,A ∈ GL(n)
• δBM(K ,Pm) = minδBM(K ,P) : P ∈ Pm
• Ωn(K ) centro-affine surface area of K
Gruber 1993: ∃ cn > 0
limm→∞
(δBM(K ,Pm)− 1
)m
2n−1 = cn Ωn(K )
2n−1
for K ∈ Kn, ∂K ∈ C2
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Polytopal Approximation
• Kn0 convex bodies in Rn, 0 ∈ intK
• Banach-Mazur distanceδBM(K ,P) =minλ > 1 : P ⊂ AK ⊂ λP,A ∈ GL(n)
• δBM(K ,Pm) = minδBM(K ,P) : P ∈ Pm
• Ωn(K ) centro-affine surface area of K
Gruber 1993: ∃ cn > 0
limm→∞
(δBM(K ,Pm)− 1
)m
2n−1 = cn Ωn(K )
2n−1
for K ∈ Kn, ∂K ∈ C2
![Page 42: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/42.jpg)
Polytopal Approximation
• Kn0 convex bodies in Rn, 0 ∈ intK
• Banach-Mazur distanceδBM(K ,P) =minλ > 1 : P ⊂ AK ⊂ λP,A ∈ GL(n)
• δBM(K ,Pm) = minδBM(K ,P) : P ∈ Pm• Ωn(K ) centro-affine surface area of K
Gruber 1993: ∃ cn > 0
limm→∞
(δBM(K ,Pm)− 1
)m
2n−1 = cn Ωn(K )
2n−1
for K ∈ Kn, ∂K ∈ C2
![Page 43: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/43.jpg)
Polytopal Approximation
• Choose m points i.i.d. according tothe cone measure µK on ∂K
• Pm convex hull of these m points
• Ωp(K ) Lp affine surface area of K
Reitzner 2002, Schutt & Werner 2003:∃ cn > 0
limm→∞
(Vn(K )− EVn(Pm)
)m
2n+1 = cn Ω n
n−2(K )
for K ∈ Kn0 , ∂K ∈ C2, n ≥ 3
![Page 44: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/44.jpg)
Polytopal Approximation
• Choose m points i.i.d. according tothe cone measure µK on ∂K
• Pm convex hull of these m points
• Ωp(K ) Lp affine surface area of K
Reitzner 2002, Schutt & Werner 2003:∃ cn > 0
limm→∞
(Vn(K )− EVn(Pm)
)m
2n+1 = cn Ω n
n−2(K )
for K ∈ Kn0 , ∂K ∈ C2, n ≥ 3
![Page 45: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/45.jpg)
Polytopal Approximation
• Choose m points i.i.d. according tothe cone measure µK on ∂K
• Pm convex hull of these m points
• Ωp(K ) Lp affine surface area of K
Reitzner 2002, Schutt & Werner 2003:∃ cn > 0
limm→∞
(Vn(K )− EVn(Pm)
)m
2n+1 = cn Ω n
n−2(K )
for K ∈ Kn0 , ∂K ∈ C2, n ≥ 3
![Page 46: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/46.jpg)
Lp affine surface area
• Lutwak 1996, p > 1; Hug 1996, p > 0
Ωp(K ) =
∫∂K
κ0(K , x)p
n+p dµK (x)
• Berck & Bernig & Vernicos 2008, Leng & Lu 2008, Leng &Lu & Yuan 2007, Leng & Wang 2007, Lutwak & Oliker 1995,Meyer & Werner 2000, Schutt & Werner 2004, Werner 2007,Werner & Ye 2008
• Lp Brunn Minkowski Theory:Campi, Chen, Chou, Firey, Gronchi, Haberl, Hu, Hug, Ludwig,Lutwak, Ma, Meyer, Reitzner, Schneider, Schuster, Schutt,Shen, Stancu, Umanskiy, Wang, Werner, Yang, Zhang,Zvavitch, . . .
• Bastero, Gardner, Jensen, Klain, Koldobsky, Lonke, Romance,Rubin, Ryabogin, Volcic, Yaskina, Yaskin, . . .
![Page 47: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/47.jpg)
Lp affine surface area
• Lutwak 1996, p > 1; Hug 1996, p > 0
Ωp(K ) =
∫∂K
κ0(K , x)p
n+p dµK (x)
• Berck & Bernig & Vernicos 2008, Leng & Lu 2008, Leng &Lu & Yuan 2007, Leng & Wang 2007, Lutwak & Oliker 1995,Meyer & Werner 2000, Schutt & Werner 2004, Werner 2007,Werner & Ye 2008
• Lp Brunn Minkowski Theory:Campi, Chen, Chou, Firey, Gronchi, Haberl, Hu, Hug, Ludwig,Lutwak, Ma, Meyer, Reitzner, Schneider, Schuster, Schutt,Shen, Stancu, Umanskiy, Wang, Werner, Yang, Zhang,Zvavitch, . . .
• Bastero, Gardner, Jensen, Klain, Koldobsky, Lonke, Romance,Rubin, Ryabogin, Volcic, Yaskina, Yaskin, . . .
![Page 48: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/48.jpg)
Lp affine surface area
• Lutwak 1996, p > 1; Hug 1996, p > 0
Ωp(K ) =
∫∂K
κ0(K , x)p
n+p dµK (x)
• Berck & Bernig & Vernicos 2008, Leng & Lu 2008, Leng &Lu & Yuan 2007, Leng & Wang 2007, Lutwak & Oliker 1995,Meyer & Werner 2000, Schutt & Werner 2004, Werner 2007,Werner & Ye 2008
• Lp Brunn Minkowski Theory:Campi, Chen, Chou, Firey, Gronchi, Haberl, Hu, Hug, Ludwig,Lutwak, Ma, Meyer, Reitzner, Schneider, Schuster, Schutt,Shen, Stancu, Umanskiy, Wang, Werner, Yang, Zhang,Zvavitch, . . .
• Bastero, Gardner, Jensen, Klain, Koldobsky, Lonke, Romance,Rubin, Ryabogin, Volcic, Yaskina, Yaskin, . . .
![Page 49: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/49.jpg)
Lp affine surface area
• Lutwak 1996, p > 1; Hug 1996, p > 0
Ωp(K ) =
∫∂K
κ0(K , x)p
n+p dµK (x)
• Berck & Bernig & Vernicos 2008, Leng & Lu 2008, Leng &Lu & Yuan 2007, Leng & Wang 2007, Lutwak & Oliker 1995,Meyer & Werner 2000, Schutt & Werner 2004, Werner 2007,Werner & Ye 2008
• Lp Brunn Minkowski Theory:Campi, Chen, Chou, Firey, Gronchi, Haberl, Hu, Hug, Ludwig,Lutwak, Ma, Meyer, Reitzner, Schneider, Schuster, Schutt,Shen, Stancu, Umanskiy, Wang, Werner, Yang, Zhang,Zvavitch, . . .
• Bastero, Gardner, Jensen, Klain, Koldobsky, Lonke, Romance,Rubin, Ryabogin, Volcic, Yaskina, Yaskin, . . .
![Page 50: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/50.jpg)
Properties of Ωp : Kn0 → R (Lutwak 1996)
• Ωp is SL(n) invariant and homogeneous of degreeq = n(n − p)/(n + p)
Ωp(AK ) = Ωp(K ) ∀A ∈ SL(n),∈ Rn
Ωp(λK ) = λq Ωp(K ) ∀λ > 0
• Ωp is upper semicontinuous
Ωp(K ) ≥ lim supj→∞
Ωp(Kj)
if Kj → K as j →∞.
• Ωp is a valuation
• Ω1 = Ω is translation invariant (equi-affine surface area)
• Ωn is GL(n) invariant (centro-affine surface area)
![Page 51: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/51.jpg)
Properties of Ωp : Kn0 → R (Lutwak 1996)
• Ωp is SL(n) invariant and homogeneous of degreeq = n(n − p)/(n + p)
Ωp(AK ) = Ωp(K ) ∀A ∈ SL(n),∈ Rn
Ωp(λK ) = λq Ωp(K ) ∀λ > 0
• Ωp is upper semicontinuous
Ωp(K ) ≥ lim supj→∞
Ωp(Kj)
if Kj → K as j →∞.
• Ωp is a valuation
• Ω1 = Ω is translation invariant (equi-affine surface area)
• Ωn is GL(n) invariant (centro-affine surface area)
![Page 52: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/52.jpg)
Properties of Ωp : Kn0 → R (Lutwak 1996)
• Ωp is SL(n) invariant and homogeneous of degreeq = n(n − p)/(n + p)
Ωp(AK ) = Ωp(K ) ∀A ∈ SL(n),∈ Rn
Ωp(λK ) = λq Ωp(K ) ∀λ > 0
• Ωp is upper semicontinuous
Ωp(K ) ≥ lim supj→∞
Ωp(Kj)
if Kj → K as j →∞.
• Ωp is a valuation
• Ω1 = Ω is translation invariant (equi-affine surface area)
• Ωn is GL(n) invariant (centro-affine surface area)
![Page 53: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/53.jpg)
Properties of Ωp : Kn0 → R (Lutwak 1996)
• Ωp is SL(n) invariant and homogeneous of degreeq = n(n − p)/(n + p)
Ωp(AK ) = Ωp(K ) ∀A ∈ SL(n),∈ Rn
Ωp(λK ) = λq Ωp(K ) ∀λ > 0
• Ωp is upper semicontinuous
Ωp(K ) ≥ lim supj→∞
Ωp(Kj)
if Kj → K as j →∞.
• Ωp is a valuation
• Ω1 = Ω is translation invariant (equi-affine surface area)
• Ωn is GL(n) invariant (centro-affine surface area)
![Page 54: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/54.jpg)
Properties of Ωp : Kn0 → R (Lutwak 1996)
• Ωp is SL(n) invariant and homogeneous of degreeq = n(n − p)/(n + p)
Ωp(AK ) = Ωp(K ) ∀A ∈ SL(n),∈ Rn
Ωp(λK ) = λq Ωp(K ) ∀λ > 0
• Ωp is upper semicontinuous
Ωp(K ) ≥ lim supj→∞
Ωp(Kj)
if Kj → K as j →∞.
• Ωp is a valuation
• Ω1 = Ω is translation invariant (equi-affine surface area)
• Ωn is GL(n) invariant (centro-affine surface area)
![Page 55: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/55.jpg)
SL(n) invariant valuations on Kn0
• V0(K ) Euler characteristic
• Vn(K ) volume
• Vn(K ∗) volume of K ∗
K ∗ = x ∈ Rn : x · y ≤ 1 for y ∈ K polar body of K ∈ Kn0
• Ωp(K ) Lp affine surface area
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SL(n) invariant valuations on Kn0
• V0(K ) Euler characteristic
• Vn(K ) volume
• Vn(K ∗) volume of K ∗
K ∗ = x ∈ Rn : x · y ≤ 1 for y ∈ K polar body of K ∈ Kn0
• Ωp(K ) Lp affine surface area
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Theorem (Ludwig 2002)
A functional Φ : Pn0 → R is a measurable, SL(n) invariant
valuation that is homogeneous of degree p ∈ R⇐⇒
∃ c ∈ R :
Φ(P) =
c V0(P) p = 0
c Vn(P) p = n
c Vn(P∗) p = −n0 otherwise
for every P ∈ Pn0 .
DefinitionPn0 convex polytopes in Rn, 0 ∈ intP
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Theorem (Ludwig & Reitzner 2007)
A functional Φ : Kn0 → R is an upper semicontinuous, SL(n)
invariant valuation that is homogeneous of degree q
⇐⇒
∃ c0 ∈ R, c1 ≥ 0 such that
Φ(K ) =
c0 V0(K ) + c1 Ωn(K ) for q = 0
c1 Ωp(K ) for −n < q < n and q 6= 0
c0 Vn(K ) for q = n
c0 Vn(K ∗) for q = −n0 for q < −n or q > n
for every K ∈ Kn0 where p = n(n − q)/(n + q).
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Theorem (Ludwig & Reitzner 2007)
A functional Φ : Kn0 → R is an upper semicontinuous, GL(n)
invariant valuation⇐⇒
∃ c0 ∈ R, c1 ≥ 0 such that
Φ(K ) = c0 V0(K ) + c1 Ωn(K )
for every K ∈ Kn0 .
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Theorem (Ludwig & Reitzner 2007)
A functional Φ : Kn0 → R is an upper semicontinuous, SL(n)
invariant valuation that vanishes on Pn0
⇐⇒∃φ ∈ Conc[0,∞) such that
Φ(K ) =
∫∂Kφ(κ0(K , x)) dµK (x)
for every K ∈ Kn0 .
Definitionφ ∈ Conc[0,∞) ⇐⇒
φ : [0,∞)→ [0,∞) concave, limt→0
φ(t) = limt→∞
φ(t)
t= 0
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Affine isoperimetric inequalities
• Classical equi-affine isoperimetric inequality:for K ∈ Kn and BK the centered ball s.t. Vn(BK ) = Vn(K )
Ω(K ) ≤ Ω(BK )
Blaschke 1916, Santalo 1949, Deicke 1953, Petty 1985,Leichtweiß 1988, Lutwak 1991
• Lp affine isoperimetric inequality:for K ∈ Kn
c and BK the centered ball s.t. Vn(BK ) = Vn(K )
Ωp(K ) ≤ Ωp(BK )
Lutwak 1999
Definition
Knc convex bodies in Rn with centroid at the origin 0 ∈ intK
![Page 62: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/62.jpg)
Affine isoperimetric inequalities
• Classical equi-affine isoperimetric inequality:for K ∈ Kn and BK the centered ball s.t. Vn(BK ) = Vn(K )
Ω(K ) ≤ Ω(BK )
Blaschke 1916, Santalo 1949, Deicke 1953, Petty 1985,Leichtweiß 1988, Lutwak 1991
• Lp affine isoperimetric inequality:for K ∈ Kn
c and BK the centered ball s.t. Vn(BK ) = Vn(K )
Ωp(K ) ≤ Ωp(BK )
Lutwak 1999
Definition
Knc convex bodies in Rn with centroid at the origin 0 ∈ intK
![Page 63: SL(n) invariant notions of surface area · SL(n) invariant notions of surface area Monika Ludwig Polytechnic Institute of New York University AMS Sectional Meeting, Wesleyan University,](https://reader033.fdocuments.us/reader033/viewer/2022042315/5f03f5267e708231d40b9be9/html5/thumbnails/63.jpg)
Theorem (Ludwig 2008)
Let K ∈ Knc and let BK be the centered ball s.t. Vn(BK ) = Vn(K ).
If φ ∈ Conc[0,∞), then
Ωφ(K ) ≤ Ωφ(BK )
and there is equality if K is a centered ellipsoid.
Definition
Ωφ(K ) =
∫∂Kφ(κ0(K , x)) dµK (x)
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Thank you !!!