Valuations on Convex Bodies and Sobolev Spaces

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Valuations on Convex Bodies and Sobolev Spaces

Monika Ludwig

Technische Universit¨at Wien

Szeged, May 2012

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Valuations on Convex Bodies

Kⁿ space of convex bodies (compact convex sets) in Rⁿ xA, y Abelian semigroup

A function z:KⁿÑ xA, yis a valuation ðñ zpKq zpLq zpK YLq zpK XLq for all K,LPKⁿ such thatK YLPKⁿ.

Hilbert’s Third Problem:

Dehn 1902, Sydler 1965, Jessen & Thorup 1978, McMullen 1989, . . . Classification of valuations:

Blaschke 1937,Hadwiger1949, Schneider 1971, Groemer 1972, McMullen 1977, Betke & Kneser 1985, Klain 1995, L. 1999, Reitzner 1999, Alesker 1999, Fu 2006, Haberl 2006, Schuster 2006, Tsang 2010, Wannerer 2010, Abardia 2011, Bernig & Fu 2011, Parapatits 2011, . . .

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Valuations on Convex Bodies

A function z:KⁿÑ xA, y is a valuation ðñ zpKq zpLq zpK YLq zpK XLq for all K,LPKⁿ such thatK YLPKⁿ.

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Valuations on Convex Bodies

Dehn 1902, Sydler 1965, Jessen & Thorup 1978, McMullen 1989, . . .

Classification of valuations:

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Valuations on Convex Bodies

Blaschke 1937,Hadwiger1949, Schneider 1971, Groemer 1972, McMullen 1977, Betke & Kneser 1985, Klain 1995, L.

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Hadwiger’s Classification Theorem 1952

Theorem

A functional z:KⁿÑ xR, yis a continuous and rigid motion invariant valuation

ðñ Dc₀,c₁, . . . ,c_nPR such that

zpKq c₀V₀pKq c_nV_npKq

for every K PKⁿ.

V₀pKq, . . . ,V_npKq intrinsic volumes of K

Vn n-dimensional volume

n Vn1pKq SpKq surface area

New proof by Dan Klain 1995

“Introduction to Geometric Probability” by Klain & Rota 1997

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Hadwiger’s Classification Theorem 1952

Theorem

A functional z:KⁿÑ xR, yis a continuous and rigid motion invariant valuation

ðñ Dc₀,c₁, . . . ,c_nPR such that

zpKq c₀V₀pKq c_nV_npKq

for every K PKⁿ.

V₀pKq, . . . ,V_npKq intrinsic volumes of K

Vn n-dimensional volume

p q p q

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Valuations on Convex Bodies

Real valued valuations z:KⁿÑ xR, y:

Translation invariant, continuous: McMullen, Alesker, ...

Rotation invariant, continuous: Alesker, ...

SLpnqinvariant, upper semicontinuous: L. & Reitzner, ...

Tensor valued valuations z:KⁿÑ xT^d, y:

Vector valued: Hadwiger & Schneider, ...

Symmetricd tensor valued: McMullen, Alesker, Hug & Schneider, ...

Body valued valuations:

Minkowski valuations z:KⁿÑ xKⁿ, y:

L., Schuster, Schneider, Wannerer, Abardia & Bernig, ...

L^p Minkowski valuations z:KⁿÑ xKⁿ, py: L., Haberl, Wannerer, Parapatits, ...

Radial valuations z:KⁿÑ xSⁿ,˜_py: L., Haberl, ...

Blaschke valuations z:KⁿÑ xK_cⁿ,#y: Haberl, ...

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Valuations on Convex Bodies

Blaschke valuations z:KⁿÑ xK_cⁿ,#y: Haberl, ...

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Valuations on Convex Bodies

Blaschke valuations z:KⁿÑ xKⁿ_c,#y: Haberl, ...

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Valuations on Function Spaces

F tf :X ÑRu space of real valued functions on X f _g maxtf,gu,f ^g mintf,gu

xA, y Abelian semigroup

A function z:F Ñ xA, yis a valuation ðñ zpfq zpgq zpf _gq zpf ^gq for all f,g PF such that f _g,f ^g PF.

Birkhoff: Lattice theory1940

Valuations on L^p stars: Dan Klain (AIM 1996, 1997)

Valuations on L^p spaces: Andy Tsang (IMRN 2010, TAMS 2011+) Valuations on Orlicz spaces: Hassane Kone (2011+)

Valuations on BVpRⁿq: Tuo Wang (2011+)

Valuations on Sobolev spaces: L. (AIM 2011, AJM 2012, ...)

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Valuations on Function Spaces

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Valuations on Function Spaces

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Valuations on Function Spaces

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Sobolev Spaces

W^1,ppRⁿq tf PL^ppRⁿq:|∇f| PL^ppRⁿqu p ¥1,n¥3

f _g maxtf,gu,f ^g mintf,gu W^1,ppRⁿq lattice

A function z:W^1,ppRⁿq Ñ xA, yis a valuation ðñ zpfq zpgq zpf _gq zpf ^gq for all f,g PW^1,ppRⁿq.

Question

Classification of valuations onW^1,ppRⁿq

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Sobolev Spaces

Question

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Sobolev Spaces

Question

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Sobolev Spaces

Question

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Matrix-Valued Valuations on Sobolev Spaces

W^1,2pRⁿq tf PL²pRⁿq:|∇f| PL²pRⁿqu Mⁿ space of symmetric nn matrices

Z:W^1,2pRⁿq ÑMⁿ is GLpnq contravariant ô

Zpf φq |detφ|^qφ^tZpfqφ¹ @φPGLpnq Z:W^1,2pRⁿq ÑMⁿ is affinely contravariantô

Z is GLpnq contravariant, translation invariant and homogeneous

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Matrix-Valued Valuations on Sobolev Spaces

W^1,2pRⁿq tf PL²pRⁿq:|∇f| PL²pRⁿqu

Mⁿ space of symmetric nn matrices

Zpf φq |detφ|^qφ^tZpfqφ¹ @φPGLpnq

Z:W^1,2pRⁿq ÑMⁿ is affinely contravariantô

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Matrix-Valued Valuations on Sobolev Spaces

W^1,2pRⁿq tf PL²pRⁿq:|∇f| PL²pRⁿqu

Mⁿ space of symmetric nn matrices

Zpf φq |detφ|^qφ^tZpfqφ¹ @φPGLpnq Z:W^1,2pRⁿq ÑMⁿ is affinely contravariantô

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A Characterization of the Fisher Information Matrix

Theorem (L.: AIM 2011)

A function Z:W^1,2pRⁿq Ñ xMⁿ, yis a continuous and affinely contravariant valuation

Dc PRsuch that ðñ

Zpfq cJpf²q for every f PW^1,2pRⁿq.

J_ijpgq

»

Rⁿ

Bloggpxq Bxi

Bloggpxq Bxj

gpxqdx

Connection between Fisher information matrix and LYZ ellipsoid (Lutwak, Yang & Zhang: DMJ 2000)

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A Characterization of the Fisher Information Matrix

Dc PRsuch that ðñ

J_ijpgq

»

Rⁿ

Bloggpxq Bxi

Bloggpxq Bxj

gpxqdx

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A Characterization of the Fisher Information Matrix

Dc PRsuch that ðñ

J_ijpgq

»

Rⁿ

Bloggpxq Bxi

Bloggpxq Bxj

gpxqdx

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A Characterization of the LYZ Matrix

Theorem (L.: DMJ 2003)

A function Z:P₀ⁿÑ xM, yis a valuation such that ZpφPq |detφ|φ^tZpPqφ¹ @φPGLpnq Dc PR such that ðñ

ZpPq cLpPq for every P PP₀ⁿ.

P₀ⁿ convex polytopes inRⁿ containing the origin in their interiors

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Convex-Body-Valued Valuations on Sobolev Spaces

Kⁿ_c space of origin-symmetric convex bodies

z:W^1,1pRⁿq ÑKⁿ_c is GLpnqcontravariant ô

zpf φq |detφ|^qφ^tzpfq @φPGLpnq z:W^1,1pRⁿq ÑKⁿ_c is affinely contravariantô

z is GLpnq contravariant, translation invariant and homogeneous z:W^1,1pRⁿq ÑKⁿ_c is GLpnqcovariant ô

zpf φq |detφ|^qφzpfq @φPGLpnq z:W^1,1pRⁿq ÑKⁿ_c is affinely covariantô

z is GLpnq covariant, translation invariant and homogeneous

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Convex-Body-Valued Valuations on Sobolev Spaces

Kⁿ_c space of origin-symmetric convex bodies z:W^1,1pRⁿq ÑKⁿ_c is GLpnq contravariant ô

zpf φq |detφ|^qφ^tzpfq @φPGLpnq

z:W^1,1pRⁿq ÑKⁿ_c is affinely contravariantô

z is GLpnq contravariant, translation invariant and homogeneous z:W^1,1pRⁿq ÑKⁿ_c is GLpnqcovariant ô

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Convex-Body-Valued Valuations on Sobolev Spaces

z is GLpnq contravariant, translation invariant and homogeneous

z:W^1,1pRⁿq ÑKⁿ_c is GLpnqcovariant ô

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Convex-Body-Valued Valuations on Sobolev Spaces

z is GLpnq contravariant, translation invariant and homogeneous z:W^1,1pRⁿq ÑKⁿ_c is GLpnq covariant ô

zpf φq |detφ|^qφzpfq @φPGLpnq

z:W^1,1pRⁿq ÑKⁿ_c is affinely covariantô

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Convex-Body-Valued Valuations on Sobolev Spaces

z is GLpnq contravariant, translation invariant and homogeneous z:W^1,1pRⁿq ÑKⁿ_c is GLpnq covariant ô

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A Characterization of the LYZ operator

Theorem (L.: AJM 2012)

An operator z:W^1,1pRⁿq Ñ xKⁿ_c, yis a continuous and affinely contravariant valuation

Dc ¥0such that ðñ

zpfq cΠxfy for every f PW^1,1pRⁿq.

K L tx y :x PK,y PLu Minkowski sum ofK,LPKⁿ_c xy:W^1,1pRⁿq ÑKⁿ_c,f ÞÑ xfy LYZ operator

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Origin-symmetric Convex Bodies K

ⁿ_c

Support function hpK,q:RⁿÑR

K hpK,uq

u

hpK,uq maxtux :xPKu

hpK,u vq ¤hpK,uq hpK,vq sublinear and even

Surface area measure SpK,q:BpSⁿ¹q Ñ r0,8q

K

SpK,tuuq u

SpK, ωq

Hⁿ¹ptxPbdK :npK,xq Pωuq

npK,xq outer unit normal vector toK atx PbdK

SpK,q even measure,

not concentrated on a great sphere

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Origin-symmetric Convex Bodies K

ⁿ_c

K hpK,uq

u

hpK,uq maxtux :xPKu

K

SpK,tuuq u

SpK, ωq

Hⁿ¹ptxPbdK :npK,xq Pωuq

SpK,q even measure,

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Origin-symmetric Convex Bodies K

ⁿ_c

K hpK,uq

u

hpK,uq maxtux :xPKu

K

SpK,tuuq

u SpK, ωq

Hⁿ¹ptx PbdK :npK,xq Pωuq

SpK,q even measure,

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Origin-symmetric Convex Bodies K

ⁿ_c

K hpK,uq

u

hpK,uq maxtux :xPKu

SpK,tuuq

u SpK, ωq

Hⁿ¹ptx PbdK :npK,xq Pωuq

npK,xq outer unit normal vector toK

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Projection Body, Π K , of K

u^K hyperplane orthogonal tou K|u^K projection of K to u^K V_n₁ pn1q-dimensional volume

hpΠxfy,uq

»

Rⁿ

|u∇fpxq|dx

Definition (Minkowski 1901)

hpΠK,uq V_n₁pK|u^Kq 1 2

»

Sⁿ¹

|uv|dSpK,vq

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Projection Body, Π K , of K

u^K hyperplane orthogonal tou K|u^K projection of K to u^K V_n₁ pn1q-dimensional volume hpΠxfy,uq

»

Rⁿ

|u∇fpxq|dx

Definition (Minkowski 1901)

hpΠK,uq V pK|u^Kq 1»

|uv|dSpK,vq

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Sobolev Inequality

1 n

»

Rⁿ

|∇fpxq|dx ¥vn¹^{ⁿf

n{pn1q

f PW^1,1pRⁿq tf PL¹pRⁿq,|∇f| PL¹pRⁿqu

|x| Euclidean norm of xPRⁿ f

p ³

Rⁿ|fpxq|^pdx1{p

v_n volume of n-dimensional unit ball

Equality for indicator functions of balls

Equivalent to Euclidean isoperimetric inequality Federer & Fleming 1960, Maz¹ya 1960

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Sobolev Inequality

1 n

»

Rⁿ

|∇fpxq|dx ¥vn¹^{ⁿf

n{pn1q

f PW^1,1pRⁿq tf PL¹pRⁿq,|∇f| PL¹pRⁿqu

|x| Euclidean norm of xPRⁿ f

p ³

Rⁿ|fpxq|^pdx1{p

v_n volume of n-dimensional unit ball Equality for indicator functions of balls

Equivalent to Euclidean isoperimetric inequality

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General Sobolev Inequality

1 n

»

Rⁿ

}∇fpxq}Kdx ¥vn¹^{ⁿf

n{pn1q

f PW^1,1pRⁿq

Kⁿ_c origin-symmetric convex bodies (compact convex sets) in Rⁿ K PKⁿ_c with VpKq v_n

K tx PRⁿ:xy ¤1 for all y PKu polar body ofK } }L norm with unit ball L

Equality for f 1K

Equivalent to Minkowski inequality Gromov 1986

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General Sobolev Inequality

1 n

»

Rⁿ

n{pn1q

f PW^1,1pRⁿq

Kⁿ_c origin-symmetric convex bodies (compact convex sets) in Rⁿ K PKⁿ_c with VpKq v_n

K tx PRⁿ:xy ¤1 for all y PKu polar body ofK } }L norm with unit ball L

Equality for f 1K

Equivalent to Minkowski inequality

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Optimal Sobolev Inequality

1 n

»

Rⁿ

n{pn1q

f PW^1,1pRⁿq,VpKq v_n

Question (Lutwak, Yang & Zhang 2006)

For given f PW^1,1pRⁿq, which convex body K (of volumevn) minimizes 1

n

»

Rⁿ

}∇fpxq}Kdx? Which norm is optimal?

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Optimal Sobolev Inequality

1 n

»

Rⁿ

n{pn1q

f PW^1,1pRⁿq,VpKq v_n

Question (Lutwak, Yang & Zhang 2006)

For given f PW^1,1pRⁿq, which convex body K (of volume v_n) minimizes 1 »

}∇fpxq}Kdx?

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Optimal Sobolev Body

Definition (LYZ 2006)

For f PW^1,1pRⁿq, the optimal Sobolev body, xfy, of f is defined as the unique origin-symmetric convex body such that

»

Sⁿ¹

gpuqdSpxfy,uq

»

Rⁿ

gp∇fpxqqdx for all even and positively 1-homogeneous functions g PCpRⁿq.

Theorem (LYZ 2006)

For f PW^1,1pRⁿq, the infimum over all origin-symmetric convex bodies K of volume VpKq v_n over

»

Rⁿ

}∇fpxq}Kdx is attained if and only if K is a dilate of xfy.

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Optimal Sobolev Body

Definition (LYZ 2006)

For f PW^1,1pRⁿq, the optimal Sobolev body, xfy, of f is defined as the unique origin-symmetric convex body such that

»

Sⁿ¹

gpuqdSpxfy,uq

»

Rⁿ

gp∇fpxqqdx

for all even and positively 1-homogeneous functions g PCpRⁿq. Theorem (LYZ 2006)

For f PW^1,1pRⁿq, the infimum over all origin-symmetric convex bodies K of volume VpKq v_n over

»

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Affine Sobolev inequality

Theorem (Gaoyong Zhang: JDG 1999) For f PW^1,1pRⁿq,

1 n

»

Sⁿ¹

p

»

Rⁿ

|u∇fpxq|dxqⁿdu¤ p v_n

2vn1qⁿ|f|ⁿⁿ

n1

.

Affine isoperimetric inequality H¨older’s inequalityñ

1 nv_n

»

Sⁿ¹

p

»

Rⁿ

|u∇fpxq|dxqⁿ¹

n ¤ 1

nv_n

»

Sⁿ¹

»

Rⁿ

|u∇fpxq|dx du

ñ Sobolev inequality

Left hand side is multiple ofVpΠxfyq

Extended to BVpRⁿq by Tuo Wang (AIM 2012)

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Affine Sobolev inequality

1 n

»

Sⁿ¹

p

»

Rⁿ

2vn1qⁿ|f|ⁿⁿ

n1

.

Affine isoperimetric inequality

H¨older’s inequalityñ 1

nv_n

»

Sⁿ¹

p

»

Rⁿ

|u∇fpxq|dxqⁿ¹

n ¤ 1

nv_n

»

Sⁿ¹

»

Rⁿ

|u∇fpxq|dx du

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Affine Sobolev inequality

1 n

»

Sⁿ¹

p

»

Rⁿ

2vn1qⁿ|f|ⁿⁿ

n1

.

1 nv_n

»

Sⁿ¹

p

»

Rⁿ

|u∇fpxq|dxqⁿ¹

n ¤ 1

nv_n

»

Sⁿ¹

»

Rⁿ

|u∇fpxq|dx du

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Affine Sobolev inequality

1 n

»

Sⁿ¹

p

»

Rⁿ

2vn1qⁿ|f|ⁿⁿ

n1

.

1 nv_n

»

Sⁿ¹

p

»

Rⁿ

|u∇fpxq|dxqⁿ¹

n ¤ 1

nv_n

»

Sⁿ¹

»

Rⁿ

|u∇fpxq|dx du

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Affine Sobolev inequality

1 n

»

Sⁿ¹

p

»

Rⁿ

2vn1qⁿ|f|ⁿⁿ

n1

.

1 nv_n

»

Sⁿ¹

p

»

Rⁿ

|u∇fpxq|dxqⁿ¹

n ¤ 1

nv_n

»

Sⁿ¹

»

Rⁿ

|u∇fpxq|dx du ñ Sobolev inequality

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Affine Sobolev inequality

1 n

»

Sⁿ¹

p

»

Rⁿ

2vn1qⁿ|f|ⁿⁿ

n1

.

1 nv_n

»

Sⁿ¹

p

»

Rⁿ

|u∇fpxq|dxqⁿ¹

n ¤ 1

nv_n

»

Sⁿ¹

»

Rⁿ

|u∇fpxq|dx du

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Valuations on Sobolev Spaces

xy:W^1,1pRⁿq ÑKⁿ_c is affinely covariant (LYZ 2006).

xy:W^1,1pRⁿq ÑKⁿ_c is continuous. xy:W^1,1pRⁿq Ñ xKⁿ_c,#y is a valuation.

SpK #L,q SpK,q SpL,q Blaschke addition Theorem (L. 2012)

An operator z:W^1,1pRⁿq Ñ xKⁿ_c,#yis a continuous and affinely covariant valuation

zpfq cxfy for every f PW^1,1pRⁿq.

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Valuations on Sobolev Spaces

xy:W^1,1pRⁿq ÑKⁿ_c is continuous.

xy:W^1,1pRⁿq Ñ xKⁿ_c,#y is a valuation.

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Valuations on Sobolev Spaces

SpK #L,q SpK,q SpL,q Blaschke addition

Theorem (L. 2012)

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Valuations on Sobolev Spaces

D ¥ ðñ

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Valuations on Sobolev Spaces

Πxy:W^1,1pRⁿq ÑK_cⁿis affinely contravariant (LYZ 2006).

Πxy:W^1,1pRⁿq ÑK_cⁿis continuous.

Πxy:W^1,1pRⁿq Ñ xKⁿ_c, y is a valuation.

Theorem (L. 2012)

zpfq cΠxfy for every f PW^1,1pRⁿq.

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Valuations on Sobolev Spaces

Πxy:W^1,1pRⁿq ÑK_cⁿis affinely contravariant (LYZ 2006).

Πxy:W^1,1pRⁿq ÑK_cⁿis continuous.

Πxy:W^1,1pRⁿq Ñ xKⁿ_c, y is a valuation.

Theorem (L. 2012)

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Characterization of the Projection Body Operator

Theorem (L.: JDG 2010)

An operator Z:Kⁿ₀ Ñ xKⁿ, yis a continuous valuation such that ZpφKq |detφ|φ^tZK @φPGLpnq

Dc ¥0 such that ðñ

ZpKq cΠK for every K PK₀ⁿ.

Kⁿ₀ convex bodies in Rⁿ containing the origin in their interiors

L.: AIM 2002, TAMS 2005

Schuster & Wannerer: TAMS 2012; Haberl: JEMS 2011+; Abardia & Bernig: AIM 2011; Parapatits: TAMS 2012+

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Characterization of the Projection Body Operator

Kⁿ₀ convex bodies in Rⁿ containing the origin in their interiors

Schuster & Wannerer: TAMS 2012; Haberl: JEMS 2011+; Abardia & Bernig: AIM 2011; Parapatits: TAMS 2012+

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Characterization of the Projection Body Operator

Kⁿ₀ convex bodies in Rⁿ containing the origin in their interiors L.: AIM 2002, TAMS 2005

Schuster & Wannerer: TAMS 2012; Haberl: JEMS 2011+;

Abardia & Bernig: AIM 2011; Parapatits: TAMS 2012+

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Sketch of the Proof

z:W^1,1pRⁿq Ñ xKⁿ_c, y continuous, affinely contravariant valuations

L^1,1pRⁿq W^1,1pRⁿq piecewise linear continuous functions P^1,1pRⁿq L^1,1pRⁿq ‘linear elements’

P Rⁿ

`P `P PL^1,1pRⁿq

P PP₀ⁿ

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Sketch of the Proof

L^1,1pRⁿq W^1,1pRⁿq piecewise linear continuous functions

P^1,1pRⁿq L^1,1pRⁿq ‘linear elements’

P Rⁿ

`P `P PL^1,1pRⁿq

P PP₀ⁿ

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Sketch of the Proof

L^1,1pRⁿq W^1,1pRⁿq piecewise linear continuous functions P^1,1pRⁿq L^1,1pRⁿq ‘linear elements’

P Rⁿ

`P `P PL^1,1pRⁿq

P PP₀ⁿ

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

Z:P₀ⁿ Ñ xKⁿ_c, y, ZpPq zp`_Pq GLpnq contravariant valuation onP₀ⁿ L.: JDG 2010ñ zp`_Pq cΠx`_Py ñ zpfq cΠxfy

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

g

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

g

`_P

`P x PP^1,1pRⁿq

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

g

`_P

`P x PP^1,1pRⁿq

`Q `_Q _y PP^1,1pRⁿq

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

g

`_P

`P x PP^1,1pRⁿq

f _`_Q g, f ^`_Q `_P zpfq zp`Qq zpgq zp`Pq

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

g

`_P

`P x PP^1,1pRⁿq

Z:P₀ⁿ Ñ xKⁿ_c, y, ZpPq zp`_Pq GLpnq contravariant valuation onP₀ⁿ

L.: JDG 2010ñ zp`_Pq cΠx`_Py ñ zpfq cΠxfy

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

g

`_P

`P x PP^1,1pRⁿq

Z:P₀ⁿ Ñ xKⁿ_c, y, ZpPq zp`_Pq GLpnq contravariant valuation onP₀ⁿ

ñ zpfq cΠxfy

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Sketch of the Proof, cont.

f PL^1,1pRⁿq f

g

`_P

`P x PP^1,1pRⁿq

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Valuations on Convex Bodies and Sobolev Spaces

Valuations on Convex Bodies and Sobolev Spaces

Valuations on Convex Bodies

Valuations on Convex Bodies

Valuations on Convex Bodies

Valuations on Convex Bodies

Hadwiger’s Classification Theorem 1952

Hadwiger’s Classification Theorem 1952

Valuations on Convex Bodies

Valuations on Convex Bodies

Valuations on Convex Bodies

Valuations on Function Spaces

Valuations on Function Spaces

Valuations on Function Spaces

Valuations on Function Spaces

Sobolev Spaces

Sobolev Spaces

Sobolev Spaces

Sobolev Spaces

Matrix-Valued Valuations on Sobolev Spaces

Matrix-Valued Valuations on Sobolev Spaces

Matrix-Valued Valuations on Sobolev Spaces

A Characterization of the Fisher Information Matrix

A Characterization of the Fisher Information Matrix

A Characterization of the Fisher Information Matrix

A Characterization of the LYZ Matrix

Convex-Body-Valued Valuations on Sobolev Spaces

Convex-Body-Valued Valuations on Sobolev Spaces

Convex-Body-Valued Valuations on Sobolev Spaces

Convex-Body-Valued Valuations on Sobolev Spaces

Convex-Body-Valued Valuations on Sobolev Spaces

A Characterization of the LYZ operator

Origin-symmetric Convex Bodies K

Origin-symmetric Convex Bodies K

Origin-symmetric Convex Bodies K

Origin-symmetric Convex Bodies K

Projection Body, Π K , of K

Projection Body, Π K , of K

Sobolev Inequality

Sobolev Inequality

General Sobolev Inequality

General Sobolev Inequality

Optimal Sobolev Inequality

Optimal Sobolev Inequality

Optimal Sobolev Body

Optimal Sobolev Body

Affine Sobolev inequality

Affine Sobolev inequality

Affine Sobolev inequality

Affine Sobolev inequality

Affine Sobolev inequality

Affine Sobolev inequality

Valuations on Sobolev Spaces

Valuations on Sobolev Spaces

Valuations on Sobolev Spaces

Valuations on Sobolev Spaces

Valuations on Sobolev Spaces

Valuations on Sobolev Spaces

Characterization of the Projection Body Operator

Characterization of the Projection Body Operator

Characterization of the Projection Body Operator

Sketch of the Proof

Sketch of the Proof

Sketch of the Proof

Sketch of the Proof, cont.

Sketch of the Proof, cont.

Sketch of the Proof, cont.

Sketch of the Proof, cont.

Sketch of the Proof, cont.

Sketch of the Proof, cont.

Sketch of the Proof, cont.

Sketch of the Proof, cont.

Sketch of the Proof, cont.

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