The Petty Projection Inequality and BEYOND Franz Schuster Vienna University of Technology
The Euclidean Isoperimetric Inequality: "=" only if K is a ball Petty's Projection Inequality (PPI) K |uK |u K uu Cauchy's Surface Area Formula: vol n – 1 (K | u ) du. 1 n – 1n – 1 S n – 1 S(K ) = If K , then V(K )V(K ) S(K )S(K ) nnnn 1 n n n – 1 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) Cauchy's Surface Area Formula: vol n – 1 (K | u ) du. 1 n – 1n – 1 S n – 1 S(K ) = If K , then – 1 n – 1 n nnnn vol n – 1 (K | u ) – n du S n – 1 Theorem [Petty, Proc. Conf. Convexity UO 1971]: K K n – 1 V(K )V(K ) nn S(K )S(K ) nnnn n "=" only if K is an ellipsoid If K , then
Polar Projection Bodies – The PPI Reformulated Def inition [Minkowski, 1900]: h(K,u) = max{u. x: x K} Support Function projection body K projection body K of K is defined by The h( K,u) = vol n – 1 (K | u ) Zonoids in … zonoid L is a zonoid if L = K + t for some K , t .
Radial functions (K,u) = max{ 0: u K} ( * K,u) = vol n – 1 (K | u ) – 1 projection body K projection body K of K is defined by The h( K,u) = vol n – 1 (K | u ) * K : = ( K ) * Polar projection bodiespolar * Def inition [Minkowski, 1900]: "=" only for ellipsoids V(K ) n – 1 V( * K ) V(B) n – 1 V( * B) 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]: If K , then Def inition [Dupin, 1850]: centroid body K centroid body K of K is defined by The h( K,u) = K | x. u | dx. 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 ) nV 1 (K, L ) = lim t 0 + t V(K 1 t. L) – V(K ) – nV – 1 (K, L ) = lim t 0 + t polars of zonoids all convex bodies BPCI for polars of zonoids PPI for all convex bodies zonoidsall star bodies PPI for zonoids BPCI for all star bodies The Busemann-Petty Centroid Inequality – Class Reduction Class Reduction [Lutwak, Trans. AMS 1985]: where Harmonic Radial Addition (K 1 t. L,. ) – 1 = (K,. ) – 1 + t (L,. ) – 1 Based on V 1 (K, L ) = V – 1 (L, * K ), 2 n + 1
A Proof of the BPCI – Campi & Gronchi, Adv. Math 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 K t def ined by K t = conv{x + (x) v t: x A}, t [0,1]. v A Def inition [Rogers & Shephard, 1958]:
v A Proof of the BPCI – Campi & Gronchi, Adv. Math 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 K t def ined by K t = conv{x + (x) v t: x A}, t [0,1]. Def inition [Rogers & Shephard, 1958]:
v A Proof of the BPCI – Campi & Gronchi, Adv. Math 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 K t def ined by K t = conv{x + (x) v t: x A}, t [0,1]. Def inition [Rogers & Shephard, 1958]:
Let K t be a shadow system with speed function and define Then K t is the projection of K o onto e n + 1 along e n + 1 – tv. K o = conv{(x, (x)): x A} . n + 1 KoKo A Proof of the BPCI – Campi & Gronchi, Adv. Math Proposition [Shephard, Israel J. Math. 1964]:
KoKo Let K t be a shadow system with speed function and define Then K t is the projection of K o onto e n + 1 along e n + 1 – tv. K o = conv{(x, (x)): x A} . n + 1 Proposition [Shephard, Israel J. Math. 1964]: A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
Mixed Volumes V( 1 K 1 + … + m K m ) = i 1 … i n V(K i 1,…,K i n ) A Proof of the BPCI – Campi & Gronchi, Adv. Math If K t, K 1, …, K n are shadow systems, then V(K 1,…,K n ) is convex in t, in particular V(K t ) is convex tt tt Properties of Shadow Systems: Steiner symmetrization is a special volume preserving shadow system
If K t, K 1, …, K n are shadow systems, then V(K 1,…,K n ) is convex in t, in particular V(K t ) is convex Steiner symmetrization is a special volume preserving shadow system tt tt A Proof of the BPCI – Campi & Gronchi, Adv. Math Properties of Shadow Systems: K v v v
If K t, K 1, …, K n are shadow systems, then V(K 1,…,K n ) is convex in t, in particular V(K t ) is convex Steiner symmetrization is a special volume preserving shadow system tt tt A Proof of the BPCI – Campi & Gronchi, Adv. Math Properties of Shadow Systems: K v v v S v K = K 1 2
If K t, K 1, …, K n are shadow systems, then V(K 1,…,K n ) is convex in t, in particular V(K t ) is convex Steiner symmetrization is a special volume preserving shadow system tt tt A Proof of the BPCI – Campi & Gronchi, Adv. Math Properties of Shadow Systems: KK1K1 v v v S v K = K 1 2
K = K [– x,x] dx implies V( K ) = … V([– x 1, x 1 ],…, [– x n, x n ]) dx 1 …dx n. KK A Proof of the BPCI – Campi & Gronchi, Adv. Math First step: dx K t = K [– x,x] t
implies V( K t ) = … V([– x 1, x 1 ] t,…, [– x n, x n ] t ) dx 1 …dx n. KK dx K t = K [– x,x] t First step: Second step: V( (S v K )) = V( K ) V( K 0 ) + V( K 1 ) Since V( K 0 ) = V( K ) and V( K 1 ) = V( K ) this yields V( (S v K )) V( K ) V( (S v K )) V( K ). A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002
"=" only for ellipsoids V(K ) n – 1 V( * K ) V(B) n – 1 V( * B) 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]: If K , then Lutwak, Yang, Zhang, J. Diff. Geom & 2010 S v * K * (S v K ) S v K (S v K ) and
Valuations on Convex Bodies Def inition: valuation A function : is called a valuation if (K L) + (K L) = (K ) + (L) whenever K L . 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) Minkowski valuation A map : is called a Minkowski valuation if
(K L) + (K L) = (K ) + (L) Minkowski valuation A map : is called a Minkowski valuation if Valuations on Convex Bodies Def inition: whenever K L . Trivial examples are Id and – IdExamples: is a Minkowski valuation is a Minkowski valuation
SL(n) contravariant A map : is a continuous and SL(n) contravariant Minkowski valuation if and only if Classif ication of Minkowski Valuations = c for some c 0. Theorem [Haberl, J. EMS 2011]: First such characterization results of and were obtained by Ludwig (Adv. Math. 2002; Trans. AMS 2005). oRemarks: SL(n) covariant The map : is the only non-trivial continuous SL(n) covariant Minkowski valuation. o SL(n) contravariance (AK ) = A – T (K ), A SL(n)
The Isoperimetric and the Sobolev Inequality Sobolev Inequality: If f C c ( ), then || f || 1 n n || f || 1 n n n – 1 n – 1 Notation f || p = || | f (x)| p dx 1/p Isoperimetric Inequality: V(K )V(K ) S(K )S(K ) nnnn 1 n n n – 1 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]: n n || f || 1 n n n – 1 n – 1 The aff ine Zhang – Sobolev inequality is aff ine invariant and equivalent to an extended Petty projection inequality. Remarks: || f || 1 || D u f || – n du S n – 1 1 n – 1 2 n – 1 nnnn Notation D u f : = u. f If f C c ( ), then It is stronger than the classical Sobolev inequality.
L p Sobolev Inequality If 1 < p < n and f C c ( ), then Theorem [Aubin, JDG; Talenti, AMPA; 1976]: || f || p c n, p || f || p * Notation p* :=p* := np n – p Remarks: The proof is based on Schwarz symmetrization.
Schwarz Symmetrization Def inition: distribution function The distribution function of f C c ( ) is def ined by µ f (t) = V({x : | f (x)| > t}). f (x) = sup{t > 0: µ f (t) > n ||x||}. Schwarz symmetral The Schwarz symmetral f of f is def ined by f µ f = µ f f
L p Sobolev Inequality Theorem [Aubin, JDG; Talenti, AMPA; 1976]: || f || p c n, p || f || p * Remarks: The isoperimetric inequality is the geometric core of the proof for every 1 < p < n. Notation p* :=p* := np n – p The proof is based on Schwarz symmetrization. Polya – Szegö inequality Using the Polya – Szegö inequality || f || p || f || p the proof is reduced to a 1-dimensional problem. If 1 < p < n and f C c ( ), then
Sharp Aff ine L p Sobolev Inequality Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]: c n, p || f || p * || D u f || – n du S n – 1 1 n – p The aff ine L p Sobolev inequality is aff ine invariant and stronger than the classical L p Sobolev inequality. Remarks: If 1 < p < n und f C c ( ), then an, pan, p The normalization a n,p is chosen such that || D u f || – n du S n – 1 1 n – p a n, p = || f || p.
For each p > 1 a new aff ine isoperimetric inequality is needed in the proof. Sharp Aff ine L p Sobolev Inequality Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]: If 1 < p < n und f C c ( ), then affinePólya – Szegö inequality Proof. Based on affine version of the Pólya – Szegö inequality: Remark: For all p 1 ( * ) was established by [Cianchi, LYZ, Calc. Var. PDE 2010]. || D u f || – n du S n – 1 1 n – p If 1 ≤ p < n and f C c ( ), then || D u f || – n du S n – 1 1 n – p . [Zhang, JDG 1999] & [LYZ, JDG 2002]. (*)(*) c n, p || f || p * || D u f || – n du S n – 1 1 n – p an, pan, p
Petty's Projection Inequality Revisited "=" only for ellipsoids V(K ) n – 1 V( * K ) V(B) n – 1 V( * B) Theorem [Petty, 1971]: If K , then h( K,u) = vol n – 1 (K | u ) = |u. v| dS(K,v). S n – 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 ) nV 1 (K, L ) = lim t 0 + t, p | u. v | dS p (K,v), p p h( p K,u) p = c n, p S n – 1 p Def inition [LYZ, 2000]: L p projection body p K For p > 1 and K the L p projection body p K is def ined by o L pp where the L p surface area measure S p (K,. ) is determined by p p = h(L,v) p dS p (K,v). S n – 1 p V(K + p t. L) – V(K ) p V p (K, L ) = lim t 0 + t n p L p Minkowski Addition h(K + p t. L,. ) p = h(K,. ) p + t h(L,. ) p
p | u. v | dS p (K,v), p p h( p K,u) p = c n, p S n – 1 p Def inition [LYZ, 2000]: L p projection body p K For p > 1 and K the L p projection body p K is def ined by o L pp where the L p surface area measure S p (K,. ) is determined by p p = h(L,v) p dS p (K,v). S n – 1 p V(K + p t. L) – V(K ) p V p (K, L ) = lim t 0 + t n p The L p Petty Projection Inequality "=" only for centered ellipsoids V(K ) n/p – 1 V( p K ) V(B) n/p – 1 V( p B) Theorem [LYZ, J. Diff. Geom. 2000]: ** If K , then o The proof is based on Steiner symmetrization:Remarks: S v * K * (S v K ). pp Via Class Reduction an L p BPCI was deduced from the L p PPI by LYZ (J. Diff. Geom. 2000). A direct proof of the L p BPCI using Shadow Systems was given by Campi & Gronchi (Adv. Math. 2002).
Def inition: L p Minkowski valuation, if We call : an L p Minkowski valuation, if (K L) + p (K L) = K + p L whenever K L . L p Minkowski Valuations SL(n) contravariant A map : is an SL(n) contravariant L p Minkowski valuation if and only if for all P , c 1. p P + p c 2. p P + – P = for some c 1, c 2 0. Theorem [Parapatits, 2011+; Ludwig, TAMS 2005]: oo o oo Notation denotes the set of convex polytopes containing the origin. o
Asymmetric L p Projection Bodies Def inition: where (u. v) = max{ u. v, 0}.. p K + p. p K. + – p K :=p K := (symmetric) L p projection body p K The (symmetric) L p projection body p K isRemark: ( u. v ) dS p (K,v), h( p K,u) p = a n, p S n – 1 p asymmetric L p projection body p K For p > 1 and K the asymmetric L p projection body p K is def ined by o
General L p Petty Projection Inequalities Theorem [Haberl & S., J. Diff. Geom. 2009]: If p K is the convex body def ined by p K = c 1. p K + p c 2. p K, + – then "=" only for ellipsoids centered at the origin V(K ) n/p – 1 V( p K ) V(B) n/p – 1 V( p B) ** 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
General L p Petty Projection Inequalities Theorem [Haberl & S., J. Diff. Geom. 2009]: If p K is the convex body def ined by p K = c 1. p K + p c 2. p K, + – then "=" only for ellipsoids centered at the origin V(K ) n/p – 1 V( p K ) * 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 V(B)n/p V(B)n/p V(K ) n/p – 1 V( p K ) *,*,
Asymmetric Aff ine L p Sobolev Inequality Theorem [Haberl & S., J. Funct. Anal. 2009]: Remarks: The asymmetric aff ine L p Sobolev inequality is stronger than the aff ine L p Sobolev inequality of LYZ for p > 1. The aff ine L 2 Sobolev inequality of LYZ is equivalent via an aff ine transformation to the classiscal L 2 Sobolev inequality; the asymmetric inequality is not! c n, p || f || p * || D u f || – n du S n – 1 1 n – p 2 1 p + || D u f || – n du S n – 1 1 n – p Notation D u f : = max{D u f, 0} + If 1 < p < n and f C c ( ), then
An Asymmetric Aff ine Polya – Szegö Inequality Theorem [Haberl, S. & Xiao, Math. Ann. 2011]: || D u f || – n du S n – 1 1 n – p + If p 1 and f C c ( ), then || D u f || – n du S n – 1 1 n – p + Remark: The proof uses a convexification procedure which is based on the solution of the discrete data case of the L p 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 0, Theorem d n, p,q || f || q – 1 || f || r Remarks: These sharp Gagliardo-Nirenberg inequalities interpolate between the L p Sobolev and the L p 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]: || D u f || – n du S n – 1 n – p + 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 Def inition [LYZ, 2010]: Orlicz projection body K For K the Orlicz projection body K is def ined by o Suppose that : [0, ) is convex and (0) = 0. x. u h(K,u) Normalized Cone Measure h(K,u) dS(K,u) V K ( ) = 1 nV(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 ) – 1 V( K ) V(B) – 1 V( B) Theorem [LYZ, Adv. Math. 2010]: ** If K , then o Remark: For (t) = | t | p ( (t) = max{0, t} p ) the Orlicz PPI becomes the (asymmetric) L p PPI. The proof is based on Steiner symmetrization: S v * K * (S v K ) dV(K,u) ≤ 1. h( K,x) = inf > 0: S n – 1 Def inition [LYZ, 2010]: Orlicz projection body K For K the Orlicz projection body K is def ined by o 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: Suppose that MVal SO(n) has degree n – 1 and B = B. V(K ) n – 1 V( * K ) V(B) n ? V(K ) n – 1 V( * K ) Is it true that MVal SO(n) : = MVal SO(n) : = { continuous Minkowski valuation, which is translation in- and SO(n) equivariant translation in- and SO(n) equivariant}Notation: Obstacle: In general S v * K * (S v K ). Theorem [Haberl & S., 2011+]: If n = 2 and is even, then this is true! Work in progress [Haberl & S., 2011+]: If n 3 and is „generated by a zonoid“, then this is true!