the petty projection inequality and beyond franz schuster vienna university of technology
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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
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].
DefDef inition [Rogers & Shephard, 1958]:inition [Rogers & Shephard, 1958]:
v
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].
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
A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
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]:
A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
Mixed Volumes
V(1K1 + … + mKm) = i1
…in V(Ki1
,…,Kin )
A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
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
If Kt , K1 , …, Kn are shadow systems, then
V(K1,…,Kn ) is convex in t, in particular V(Kt) is convex
Steiner symmetrization is a special volume preserving shadow system
t t
t t
A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
Properties of Shadow Systems:Properties of Shadow Systems:
K
v
v
If Kt , K1 , …, Kn are shadow systems, then
V(K1,…,Kn ) is convex in t, in particular V(Kt) is convex
Steiner symmetrization is a special volume preserving shadow system
t t
t t
A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
Properties of Shadow Systems:Properties of Shadow Systems:
K
v
v
Sv K = K 1
2
If Kt , K1 , …, Kn are shadow systems, then
V(K1,…,Kn ) is convex in t, in particular V(Kt) is convex
Steiner symmetrization is a special volume preserving shadow system
t t
t t
A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
Properties of Shadow Systems:Properties of Shadow Systems:
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
A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
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 )).
A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
"=" only for ellipsoids
V(K ) n – 1V(*K ) V(B)
n – 1V(*B)
Theorem [Petty, 1971]:Theorem [Petty, 1971]: If K , then
PPI and BPCI
"=" only for centered ellipsoids
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
DefDef inition:inition:
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.
If 1 < p < n and f Cc ( ), then
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
"=" only for ellipsoids
V(K ) n – 1V(*K ) V(B)
n – 1V(*B)
Theorem [Petty, 1971]:Theorem [Petty, 1971]: If K , then
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
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
The Lp Petty Projection Inequality
"=" only for centered ellipsoids
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).
DefDef inition:inition:
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
DefDef inition:inition:
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)* *
Theorem [Haberl & S., J. Diff. Geom. 2009]:Theorem [Haberl & S., J. Diff. Geom. 2009]:
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
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 ) *
Theorem [Haberl & S., J. Diff. Geom. 2009]:Theorem [Haberl & S., J. Diff. Geom. 2009]:
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}+
If 1 < p < n and f Cc ( ), then
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
DefDef inition [LYZ, 2010]:inition [LYZ, 2010]:
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
"=" only for centered ellipsoids
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
DefDef inition [LYZ, 2010]:inition [LYZ, 2010]:
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!