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

2. The Euclidean Isoperimetric Inequality: "=" only if K is a ball Petty's Projection Inequality (PPI) K |uK |u K uu Cauchy's Surface Area Formula:  vol n – 1 (K | u  ) du. 1 n – 1n – 1 S n – 1 S(K ) = If K , then V(K )V(K )  S(K )S(K ) nnnn 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

3. 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 – 1n – 1 S n – 1 S(K ) = If K , then – 1  n – 1 n nnnn  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 ) nn   S(K )S(K ) nnnn n "=" only if K is an ellipsoid If K , then

4. 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 .

5. 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

6. 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).

7. 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

8. 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 K t def ined by K t = conv{x +  (x) v t: x  A}, t  [0,1]. v A Def inition [Rogers & Shephard, 1958]:

9. 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 K t def ined by K t = conv{x +  (x) v t: x  A}, t  [0,1]. Def inition [Rogers & Shephard, 1958]:

10. 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 K t def ined by K t = conv{x +  (x) v t: x  A}, t  [0,1]. Def inition [Rogers & Shephard, 1958]:

11. 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. 2002 Proposition [Shephard, Israel J. Math. 1964]:

12. 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

13. 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. 2002 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

14. 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. 2002 Properties of Shadow Systems: K v v  v

15. 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. 2002 Properties of Shadow Systems: K v v  v S v K = K 1 2

16. 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. 2002 Properties of Shadow Systems: KK1K1 v v  v S v K = K 1 2

17.  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. 2002 First step: dx  K t =  K [– x,x] t

18. 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 ) 1 2 1 2 1 2 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

19. "=" 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. 2000 & 2010 S v  * K   * (S v K ) S v  K   (S v K ) and

20. 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

21.  (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

22. 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)

23. 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 ) nnnn 1 n n n – 1 n – 1 [Federer & Fleming, Ann. Math. 1960] [Maz‘ya, Dokl. Akad. Nauk SSSR 1960]

24. 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 nnnn Notation D u f : = u.  f If f  C c ( ), then  It is stronger than the classical Sobolev inequality.

25. 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.

26. 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

27. 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 

28. 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.

29. 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 

30. 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 – 1 1 2 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

31. 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).

32. 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

33. Asymmetric L p Projection Bodies Def inition: where (u. v)  = max{  u. v, 0}..  p K + p.  p K. + – 1 2 1 2 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  

34. 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 

35. 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 ) *,*,

36. 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 

37. 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].

38. [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)

39. 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 )

40. 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!

41. 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!