The Petty Projection Inequality and BEYOND Franz Schuster Vienna University of Technology.

The Petty Projection Inequality

and BEYOND

Franz Schuster

Vienna University of Technology

The Euclidean Isoperimetric Inequality:The Euclidean Isoperimetric Inequality:

"=" only if K is a ball

Petty's Projection Inequality (PPI)

K |u

K

u

Cauchy's Surface Area Formula:Cauchy's Surface Area Formula:

voln – 1(K | u) du.

1n – 1 S

n – 1

S(K ) =

If K , then

V(K )S(K ) nn

1 n n

n – 1

Notation

S(K ) … Surface area of K

V(K ) … Volume of K

n … Volume of unit ball B

The following functional on is SL(n) invariant

Petty's Projection Inequality (PPI)

K |u

K

u

Cauchy's Surface Area Formula:Cauchy's Surface Area Formula:

voln – 1(K | u) du.

1n – 1 S

n – 1

S(K ) =

If K , then

– 1n – 1 n

nn voln – 1(K | u) –

ndu

S n – 1

Theorem [Petty, Proc. Conf. Convexity UO 1971]:Theorem [Petty, Proc. Conf. Convexity UO 1971]:

K

n – 1V(K )n

S(K )nn

n

"=" only if K is an ellipsoid

If K , then

Polar Projection Bodies – The PPI Reformulated

DefDef inition [Minkowski, inition [Minkowski, 1900]: 1900]:

h(K,u) = max{u . x: x K}

Support Function

projection bodyprojection body KK of K is defined byThe

h( K,u) = voln – 1(K | u)

Zonoids in …

L is a zonoidzonoid if L = K + t for some K , t .

Radial functions

(K,u) = max{ 0: u K}

(* K,u) = voln – 1(K | u) – 1

projection bodyprojection body KK of K is defined byThe

h( K,u) = voln – 1(K | u)

*K := ( K )*

Polar projection bodies

polarpolar **

DefDef inition [Minkowski, inition [Minkowski, 1900]: 1900]:

"=" only for ellipsoids

V(K ) n – 1V(*K ) V(B)

n – 1V(*B)

Theorem [Petty, 1971]:Theorem [Petty, 1971]: If K , then

Polar Projection Bodies – The PPI Reformulated

The Busemann-Petty Centroid Inequality – Class Reduction

"=" only for centered ellipsoids

V(K ) – (n + 1)V(K ) V(B) – (n + 1)V(B)

Theorem [Petty, Pacific J. Math. 1961]:Theorem [Petty, Pacific J. Math. 1961]:

If K , then

DefDef inition [Dupin, inition [Dupin, 1850]: 1850]:

centroid bodycentroid body KK of K is defined byThe

h( K,u) =

K | x . u | dx.

Remarks:Remarks:

Petty deduced the PPI from the BPCI!

The BPCI is a reformulation of the Random-Simplex Inequality by Busemann (Pacific J. Math. 1953).

V(K + t L) – V(K )nV1(K , L ) = limt 0+ t

V(K 1 t . L) – V(K )– nV– 1(K , L ) = limt 0+ t

~

BPCI for polars of zonoidspolars of zonoids PPI for all convex bodiesall convex bodies

PPI for zonoidszonoids BPCI for all star bodiesall star bodies

The Busemann-Petty Centroid Inequality – Class Reduction

Class Reduction [Lutwak, Trans. AMS 1985]:Class Reduction [Lutwak, Trans. AMS 1985]:

where

Harmonic Radial Addition

(K 1 t . L, . ) – 1 = (K, . ) – 1 + t (L, . ) – 1~

Based on

VV11((KK ,, LL ) = ) = VV–– 11((LL,, **K K ),),22nn + 1 + 1

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002

Let A be compact, a bounded function on A and let v S

n – 1. A shadow system along the direction v is a family of convex bodies Kt def ined by

Kt = conv{x + (x) v t: x A}, t [0,1].

vA

DefDef inition [Rogers & Shephard, 1958]:inition [Rogers & Shephard, 1958]:

v


Let A be compact, a bounded function on A and let v S

n – 1. A shadow system along the direction v is a family of convex bodies Kt def ined by

Kt = conv{x + (x) v t: x A}, t [0,1].

DefDef inition [Rogers & Shephard, 1958]:inition [Rogers & Shephard, 1958]:

Let Kt be a shadow system with speed function and define

Then Kt is the projection of Ko onto en + 1 along en + 1 – tv.

Ko = conv{(x,(x)): x A} . n + 1

Ko


Proposition [Shephard, Israel J. Math. 1964]:Proposition [Shephard, Israel J. Math. 1964]:

Ko

Let Kt be a shadow system with speed function and define

Then Kt is the projection of Ko onto en + 1 along en + 1 – tv.

Ko = conv{(x,(x)): x A} . n + 1

Proposition [Shephard, Israel J. Math. 1964]:Proposition [Shephard, Israel J. Math. 1964]:


Mixed Volumes

V(1K1 + … + mKm) = i1

…in V(Ki1

,…,Kin )


If Kt , K1 , …, Kn are shadow systems, then

V(K1,…,Kn ) is convex in t, in particular V(Kt) is convex

t t

t t

Properties of Shadow Systems:Properties of Shadow Systems:

Steiner symmetrization is a special volume preserving shadow system




t t

t t



K

v

v




t t

t t



K

v

v

Sv K = K 1

2




t t

t t



K K1

v

v

Sv K = K 1

2

K = K [– x,x] dx

implies

V(K ) = … V([– x1, x1],…, [– xn , xn]) dx1…dxn.K K


First stepFirst step::

dxKt =

K [– x,x]t

implies

V(Kt ) = … V([– x1, x1]t ,…, [– xn , xn]t ) dx1…dxn.K K

dxKt =

K [– x,x]t

First stepFirst step::

Second stepSecond step::

V((Sv K )) = V(K ) V(K0) + V(K1) 1

2

12

12

Since V(K0) = V(K ) and V(K1) = V(K ) this yields

VV((((SSvv KK )) )) VV((KK )).



V(K ) n – 1V(*K ) V(B)

n – 1V(*B)


PPI and BPCI


V(K ) – (n + 1)V(K ) V(B) – (n + 1)V(B)

Theorem [Busemann-Petty, 1961]:Theorem [Busemann-Petty, 1961]:If K , then

Lutwak, Yang, Zhang, J. Diff. Geom. 2000 & 2010

Sv *K *(Sv K ) Sv K (Sv K ) and

Valuations on Convex Bodies

DefDef inition:inition:

A function : is called a valuationvaluation if

(K L) + (K L) = (K ) + (L)

whenever K L .

The Theory of Valuations:The Theory of Valuations:

Abardia, Alesker, Bernig, Fu, Goodey, Groemer, Haberl, Hadwiger, Hug, Ludwig, Klain, McMullen, Parapatits, Reitzner, Schneider, Wannerer, Weil, …

(K L) + (K L) = (K ) + (L)

A map : is called a Minkowski valuationMinkowski valuation if

(K L) + (K L) = (K ) + (L)

A map : is called a Minkowski valuationMinkowski valuation if

Valuations on Convex Bodies


whenever K L .

Trivial examples are Id and – Id

Examples:Examples:

is a Minkowski valuation

is a Minkowski valuation

A map : is a continuous and SL(SL(nn) contravariant) contravariant Minkowski valuation if and only if

Classif ication of Minkowski Valuations

= c

for some c 0.

Theorem [Haberl, J. EMS 2011]:Theorem [Haberl, J. EMS 2011]:

First such characterization results of and were obtained by Ludwig (Adv. Math. 2002; Trans. AMS 2005).

o

Remarks:Remarks:

The map : is the only non-trivial continuous SL(SL(nn) covariant ) covariant Minkowski valuation.

o

SL(n) contravariance

(AK ) = A – T(K ), A SL(n)

The Isoperimetric and the Sobolev Inequality

Sobolev Inequality:Sobolev Inequality:

If f Cc ( ), then

|| f ||1 nn || f ||

1 n

n n – 1

Notation

f || p = ||

| f (x)| p dx 1/p

Isoperimetric Inequality:Isoperimetric Inequality:

V(K )S(K ) nn

1 n n

n – 1

[Federer & Fleming, Ann. Math. 1960]

[Maz‘ya, Dokl. Akad. Nauk SSSR 1960]

Aff ine Zhang – Sobolev Inequality

Theorem [Zhang, J. Diff. Geom. 1999]:Theorem [Zhang, J. Diff. Geom. 1999]:

nn || f || 1 n

n n – 1

The aff ine Zhang – Sobolev inequality is aff ine invariant and equivalent to an extended Petty projection inequality.

Remarks:Remarks:

|| f ||1 || Du f ||

– ndu

S n – 1

1 n –

12n – 1

nn

Notation

Du f := u . f

If f Cc ( ), then

It is stronger than the classical Sobolev inequality.

Lp Sobolev Inequality

If 1 < p < n and f Cc ( ), then

Theorem [Aubin, JDG; Talenti, AMPA; 1976]:Theorem [Aubin, JDG; Talenti, AMPA; 1976]:

|| f || p cn, p || f || p*

Notation

p* := np

n – p

Remarks:Remarks:

The proof is based on Schwarz symmetrization.

Schwarz Symmetrization

DefDef inition:inition:The distribution functiondistribution function of f Cc ( ) is def ined by

µf (t) = V({x : | f (x)| > t}).

f (x) = sup{t > 0: µf (t) > n ||x||}.

The Schwarz symmetralSchwarz symmetral f of f is def ined by

f

µf = µf

f

Lp Sobolev Inequality

Theorem [Aubin, JDG; Talenti, AMPA; 1976]:Theorem [Aubin, JDG; Talenti, AMPA; 1976]:

|| f || p cn, p || f || p*

Remarks:Remarks:

The isoperimetric inequality is the geometric core of the proof for every 1 < p < n.

Notation

p* := np

n – p

The proof is based on Schwarz symmetrization. Using the PolyaPolya –– Szegö inequalitySzegö inequality

||f || p ||f || p

the proof is reduced to a 1-dimensional problem.


Sharp Aff ine Lp Sobolev Inequality

Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]:Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]:

cn, p || f || p* || Du f ||

– ndu

S n – 1

1 n –

p

The aff ine Lp Sobolev inequality is aff ine invariant andstronger than the classical Lp Sobolev inequality.

Remarks:Remarks:

If 1 < p < n und f Cc ( ), then

an, p

The normalization an,p is chosen such that

|| Du f || – n

duS

n – 1

1 n –

pan, p = ||f || p .

For each p > 1 a new aff ine isoperimetric inequality is needed in the proof.

Sharp Aff ine Lp Sobolev Inequality

Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]:Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]:

If 1 < p < n und f Cc ( ), then

Proof. Based on affineaffine version of the PólyaPólya –– Szegö inequalitySzegö inequality:

Remark:Remark:For all p 1 (*) was established by

[Cianchi, LYZ, Calc. Var. PDE 2010].

|| Du f || – n du

S n – 1

1 n –

p

If 1 ≤ p < n and f Cc ( ), then

|| Du f || –

n du

S n – 1

1 n –

p .

[Zhang, JDG 1999] & [LYZ, JDG 2002].

(*)

cn, p || f || p* || Du f ||

– ndu

S n – 1

1 n –

pan, p

Petty's Projection Inequality Revisited


V(K ) n – 1V(*K ) V(B)

n – 1V(*B)


h( K,u) = voln – 1(K | u) = |u . v| dS(K,v).S n – 1

12

Cauchy‘s Projection Formula:Cauchy‘s Projection Formula: If K , then

where the surface area measure S(K, . ) is determined by

= h(L,v) dS(K,v).S n – 1

V(K + t L) – V(K )nV1(K , L ) = limt 0+ t

,

| u . v | dSpp(K,v),h(pp K,u) pp = cn, p

S n – 1

pp

DefDef inition [LYZ, 2000]:inition [LYZ, 2000]:

For p > 1 and K the LLpp projection bodyprojection body pp KK is def ined byo

where the LLpp surface area measure Spp(K, . ) is determined by

= h(L,v) pp

dSpp(K,v).

S n – 1

V(K +pp t . L) – V(K ) Vpp(K , L ) = limt 0+ tnpp

Lp Minkowski Addition

h(K +p t . L, . ) p = h(K, . ) p + t h(L, . ) p

| u . v | dSpp(K,v),h(pp K,u) pp = cn, p

S n – 1

pp


For p > 1 and K the LLpp projection bodyprojection body pp KK is def ined byo

where the LLpp surface area measure Spp(K, . ) is determined by

= h(L,v) pp

dSpp(K,v).

S n – 1

V(K +pp t . L) – V(K ) Vpp(K , L ) = limt 0+ tnpp

The Lp Petty Projection Inequality


V(K )n/p – 1V(p K ) V(B)n/p – 1V(p B)

Theorem [LYZ, J. Diff. Geom. 2000]:Theorem [LYZ, J. Diff. Geom. 2000]:

* *

If K , theno

The proof is based on Steiner symmetrization:

Remarks:Remarks:

SSv v * * KK * * ((SSv v KK ). ). pp pp

Via Class Reduction an Lp BPCI was deduced from the Lp PPI by LYZ (J. Diff. Geom. 2000). A direct proof of the Lp BPCI using Shadow Systems was given by Campi & Gronchi (Adv. Math. 2002).


We call : an LLpp Minkowski valuation Minkowski valuation, , ifif

(K L) +p (K L) = K +p L

whenever K L .

Lp Minkowski Valuations

A map : is an SL(SL(nn) contravariant) contravariant Lp Minkowski valuation if and only if for all P ,

c1 . p P +p c2 . p P+ –P =

for some c1, c2 0.

Theorem [Parapatits, 2011+; Ludwig, TAMS 2005]:Theorem [Parapatits, 2011+; Ludwig, TAMS 2005]:

o o

o

o o

Notation

denotes the set of convex polytopes containing the origin.

o

Asymmetric Lp Projection Bodies


where (u . v) = max{ u . v, 0}.

. p K +p . p K .+ –1

212

p K :=

The (symmetric)(symmetric) LLpp projection body projection body pp KK is

Remark:Remark:

( u . v ) dSp(K,v),h(p K,u)

p = an, pS

n – 1

p

For p > 1 and K the asymmetricasymmetric LLpp projection bodyprojection body pp KK

is def ined byo

General Lp Petty Projection Inequalities

Theorem [Haberl & S., J. Diff. Geom. 2009]:Theorem [Haberl & S., J. Diff. Geom. 2009]:

If p K is the convex body def ined by

p K = c1 . p K +p c2 . p K,+ –

then

"=" only for ellipsoids centered at the origin

V(K )n/p – 1V(p K ) V(B)n/p – 1V(p B)* *


If p B = B, then

V(p K ) V(p K ) V(p K )* *,*

"=" only if p = p"=" only if p = p

General Lp Petty Projection Inequalities


If p K is the convex body def ined by

p K = c1 . p K +p c2 . p K,+ –

then

"=" only for ellipsoids centered at the origin

V(K )n/p – 1V(p K ) *


If p B = B, then

V(p K ) V(p K ) V(p K )* *,*

"=" only if p = p"=" only if p = p

V(B)n/pV(K )n/p – 1V(p K )*,

Asymmetric Aff ine Lp Sobolev Inequality

Theorem [Haberl & S., J. Funct. Anal. 2009]:Theorem [Haberl & S., J. Funct. Anal. 2009]:

Remarks:Remarks:

The asymmetric aff ine Lp Sobolev inequality is stronger than the aff ine Lp Sobolev inequality of LYZ for p > 1.

The aff ine L2 Sobolev inequality of LYZ is equivalent via an aff ine transformation to the classiscal L2 Sobolev inequality; the asymmetric inequality is not!

cn, p || f || p*

|| Du f || –

n duS

n – 1

1 n –

p2

1 p +

|| Du f || –

n du

S n – 1

1 n –

p

Notation

Du f := max{Du f , 0}+


An Asymmetric Aff ine Polya – Szegö Inequality

Theorem [Haberl, S. & Xiao, Math. Ann. 2011]:Theorem [Haberl, S. & Xiao, Math. Ann. 2011]:

|| Du f || – n du

S n – 1

1 n –

p+

If p 1 and f Cc ( ), then

|| Du f || –

n du

S n – 1

1 n –

p+

Remark:Remark:

The proof uses a convexification procedure which is based on the solution of the discrete data case of the Lp Minkowski problem [Chou & Wang, Adv. Math. 2006].

[Haberl, S. & Xiao, Math. Ann. 2011]:[Haberl, S. & Xiao, Math. Ann. 2011]:

Sharp Affine Gagliardo-Nirenberg Inequalities

If 1 < p < n, p < q < p(n – 1)/(n – p) and f Cc ( ), then for suitable r( p,q), (n,p,q) > 0,

TheoremTheorem

dn, p,q || f || q – 1 || f || r

Remarks:Remarks:

These sharp Gagliardo-Nirenberg inequalities interpolate between the Lp Sobolev and the Lp logarithmic Sobolev inequalities (Del Pino & Dolbeault, J. Funct. Anal 2003).

A proof using a mass-transportation approach was given by Cordero-Erausquin, Nazaret, Villani (Adv. Math. 2004)

|| f || p

[Del Pino & Dolbeault, JMPA 2002]:[Del Pino & Dolbeault, JMPA 2002]:

|| Du f || – n du

S n – 1

n –

p+

Other Affine Analytic Inequalities include …Other Affine Analytic Inequalities include …

Affine (Asymmetric) Log-Sobolev Inequalities

Haberl, Xiao, S. (Math. Ann. '11)

Affine Moser-Trudinger and Morrey-Sobolev Inequalities

Cianchi, LYZ (Calc. Var. PDE '10)

The Orlicz-Petty Projection Inequality

dV(K,u) ≤

1 .

h( K,x) = inf > 0:S

n – 1


For K the OrliczOrlicz projection bodyprojection body KK is def ined byo

Suppose that : [0,) is convex and (0) = 0.

x . u h(K,u)

Normalized Cone Measure

h(K,u) dS(K,u)VK () =

1nV(K )

An Orlicz BPCI was also established by LYZ (J. Diff. Geom. 2010) and later by Paouris & Pivovarov.

The Orlicz-Petty Projection Inequality


V(K )– 1V( K ) V(B)– 1V( B)

Theorem [LYZ, Adv. Math. 2010]:Theorem [LYZ, Adv. Math. 2010]:

* *

If K , theno

Remark:Remark: For (t) = | t | p ((t) = max{0, t} p) the Orlicz PPI becomes the (asymmetric) Lp PPI.The proof is based on Steiner symmetrization:

SSv v * * KK * * ((SSv v KK ) )

dV(K,u) ≤

1 .

h( K,x) = inf > 0:S

n – 1


For K the OrliczOrlicz projection bodyprojection body KK is def ined byo

Suppose that : [0,) is convex and (0) = 0.

x . u h(K,u)

However, NO CLASS REDUCTION!

Open Problem – How strong is the PPI really?

Question:Question:

Suppose that MValSO(n) has degree n – 1 and B = B.

V(K )n – 1V(* K ) V(B)n ?V(K )n – 1V(*K )

Is it true that

MValMValSO(SO(nn)) ::= = { continuous Minkowski valuation, which is translation in- and SO(translation in- and SO(nn) equivariant) equivariant}

Notation:Notation:Obstacle:Obstacle:

In general

SSv v * * KK **((SSv v KK ). ).

Theorem [Haberl & S., 2011+]:Theorem [Haberl & S., 2011+]:

If n = 2 and is even, then this is true!

Work in progress [Haberl & S., 2011+]:Work in progress [Haberl & S., 2011+]:

If n 3 and is „generated by a zonoid“, then this is true!

The Petty Projection Inequality and BEYOND Franz Schuster Vienna University of Technology.

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