1 / 42

The Petty Projection Inequality and BEYOND

The Petty Projection Inequality and BEYOND. Franz Schuster Vienna University of Technology. n – 1. n. Notation. S ( K ) … Surface area of K V ( K ) … Volume of K  n … Volume of unit ball B. S ( K ). V ( K ). . Petty's Projection Inequality (PPI).

dscholz
Download Presentation

The Petty Projection Inequality and BEYOND

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. n–1 n Notation S(K) … Surface area of K V(K) … Volume of K n… Volume of unit ball B S(K) V(K)  Petty's Projection Inequality (PPI) The Euclidean Isoperimetric Inequality: 1 nn n "=" only if K is a ball K Cauchy's Surface Area Formula: If K  , then u 1  voln – 1(K|u) S(K) = du. n–1 K|u Sn – 1

  3. Theorem [Petty, Proc. Conf. Convexity UO 1971]: If K  , then The following functional on is SL(n) invariant – n – 1 1 n n – 1  voln – 1(K|u)–n K   du nn Sn – 1 S(K) n nn V(K)  n Petty's Projection Inequality (PPI) "=" only if K is an ellipsoid K Cauchy's Surface Area Formula: If K  , then u 1  voln – 1(K|u) S(K) = du. n–1 K|u Sn – 1

  4. L  is a zonoidifL = K + tforsomeK  , t . Support Function h(K,u) = max{u .x: xK} Polar Projection Bodies – The PPI Reformulated Definition [Minkowski,  1900]: The projectionbodyKofKisdefinedby h(K,u) = voln – 1(K|u) Zonoids in …

  5. Theorem [Petty, 1971]: If K  , then V(K)n – 1V(*K)  V(B)n – 1V(*B) "=" only for ellipsoids Radial functions (K,u) = max{  0: uK} Polar projection bodies *K:= (K)* Polar Projection Bodies – The PPI Reformulated Definition [Minkowski,  1900]: * The projection bodyK of K is defined by polar h(K,u) = voln – 1(K|u) (*K,u) = voln – 1(K|u) – 1

  6. If K  , then Petty deduced the PPI from the BPCI! The BPCI is a reformulation of the Random-Simplex Inequality by Busemann (Pacific J. Math. 1953). The Busemann-Petty Centroid Inequality – Class Reduction Definition [Dupin,  1850]: centroidbodyKofKisdefinedby The K|x.u|dx. h(K,u) = Theorem [Petty, Pacific J. Math. 1961]: Remarks: V(K) – (n + 1)V(K)  V(B) – (n + 1)V(B) "=" only for centered ellipsoids

  7. The Busemann-Petty Centroid Inequality – Class Reduction Class Reduction [Lutwak, Trans. AMS 1985]: BPCI forpolarsofzonoids PPI forall convexbodies PPI forzonoids  BPCI forall starbodies Based on V1(K,L) = V–1(L,*K ), where 2 n + 1 V(K + tL)–V(K) nV1(K,L) =lim t t 0+ Harmonic Radial Addition V(K 1 t .L)–V(K) –nV–1(K,L) =lim (K1t.L,.) – 1 = (K,.) – 1 + t(L,.)– 1 t t 0+

  8. v A A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Definition [Rogers & Shephard, 1958]: Let A  be compact,  a bounded function on A and let v  Sn – 1. A shadow system along the direction v is a family of convex bodies Kt defined by Kt = conv{x + (x)vt: x  A}, t  [0,1].

  9. v A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Definition [Rogers & Shephard, 1958]: Let A  be compact,  a bounded function on A and let v  Sn – 1. A shadow system along the direction v is a family of convex bodies Kt defined by Kt = conv{x + (x)vt: x  A}, t  [0,1].

  10. v A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Definition [Rogers & Shephard, 1958]: Let A  be compact,  a bounded function on A and let v  Sn – 1. A shadow system along the direction v is a family of convex bodies Kt defined by Kt = conv{x + (x)vt: x  A}, t  [0,1].

  11. A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Proposition [Shephard, Israel J. Math. 1964]: Let Kt be a shadow system with speed function  and define Ko = conv{(x,(x)): x  A}  . n + 1 Then Kt is the projection of Ko onto en + 1 along en + 1 – tv.  Ko

  12. A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Proposition [Shephard, Israel J. Math. 1964]: Let Kt be a shadow system with speed function  and define Ko = conv{(x,(x)): x  A}  . n + 1 Then Kt is the projection of Ko onto en + 1 along en + 1 – tv.  Ko

  13. A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Properties ofShadow Systems: If Kt, K1, …, Kn are shadow systems, then t t V(K1,…,Kn) isconvex in t, in particularV(Kt) isconvex t t Steiner symmetrizationis a specialvolumepreservingshadowsystem Mixed Volumes V(1K1 + … + mKm) =  i1…inV(Ki1,…,Kin )

  14. K v v A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Properties ofShadow Systems: If Kt, K1, …, Kn are shadow systems, then t t V(K1,…,Kn) isconvex in t, in particularV(Kt) isconvex t t Steiner symmetrizationis a specialvolumepreservingshadowsystem

  15. SvK = K 1 2 A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Properties ofShadow Systems: If Kt, K1, …, Kn are shadow systems, then t t V(K1,…,Kn) isconvex in t, in particularV(Kt) isconvex t t Steiner symmetrizationis a specialvolumepreservingshadowsystem K v v

  16. K1 1 2 A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Properties ofShadow Systems: If Kt, K1, …, Kn are shadow systems, then t t V(K1,…,Kn) isconvex in t, in particularV(Kt) isconvex t t Steiner symmetrizationis a specialvolumepreservingshadowsystem K SvK = K v v

  17. implies V(K) =  … V([– x1,x1],…, [– xn,xn])dx1…dxn. K K A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 First step: K = K [– x,x]dx Kt = K [– x,x] dx t

  18. 1 1 2 2 Since V(K0) = V(K) and V(K1) = V(K) this yields 1 V((SvK))  V(K). 2 A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 First step: Kt = K [– x,x] dx t implies V(Kt) =  … V([– x1,x1]t,…, [– xn,xn]t)dx1…dxn. K K Second step: V((SvK)) = V(K)  V(K0) + V(K1)

  19. Theorem [Petty, 1971]: If K  , then V(K)n – 1V(*K)  V(B)n – 1V(*B) "=" only for ellipsoids PPI and BPCI Lutwak, Yang, Zhang, J.Diff.Geom.2000 & 2010 Theorem [Busemann-Petty, 1961]: and Sv K (Sv K) Sv *K *(Sv K) V(K) – (n + 1)V(K)  V(B) – (n + 1)V(B) If K  , then "=" only for centered ellipsoids

  20. The TheoryofValuations: Abardia, Alesker, Bernig, Fu, Goodey, Groemer, Haberl, Hadwiger, Hug, Ludwig, Klain, McMullen, Parapatits, Reitzner, Schneider, Wannerer, Weil, … Valuations on Convex Bodies Definition: A function :  is called avaluation if A map :  is called a Minkowski valuation if (K  L) + (K  L) = (K) + (L) (K  L) + (K  L) = (K) + (L) whenever K  L  .

  21. Valuations on Convex Bodies Definition: A map :  iscalled a Minkowskivaluationif (K  L) + (K  L) = (K) + (L) whenever K  L  . Examples: Trivial examples are Id and –Id  is a Minkowski valuation  is a Minkowski valuation

  22. The map : istheonly non-trivial continuousSL(n) covariantMinkowskivaluation. Classification of Minkowski Valuations Theorem [Haberl, J. EMS 2011]: A map :  is a continuousandSL(n) contravariantMinkowskivaluationifandonlyif o  = c for some c  0. SL(n) contravariance (AK) = A–T(K), ASL(n) Remarks: o First such characterization results of  and  were obtained by Ludwig (Adv. Math. 2002; Trans. AMS 2005).

  23. n–1 n n–1 n IsoperimetricInequality: 1/p pdx S(K) V(K)  The Isoperimetric and the Sobolev Inequality Sobolev Inequality:  Iff Cc( ), then nn 1 ||f ||1  nn|| f || n [Federer & Fleming, Ann. Math. 1960] [Maz‘ya, Dokl. Akad. Nauk SSSR 1960] Notation f ||p=  || |f(x)| 1 n

  24. n n–1 Notation Duf:=u .f 1 – n It is stronger than the classical Sobolev inequality. Affine Zhang–Sobolev Inequality Theorem [Zhang, J. Diff. Geom. 1999]:  Iff Cc( ), then ||Du f ||–n nn 1 ||f ||1   nn|| f || du n 1 2n – 1 Sn – 1 Remarks: The affine Zhang–Sobolev inequality is affine invariant and equivalent to an extended Petty projection inequality.

  25. Lp Sobolev Inequality Notation np p*:= n – p Remarks: The proof is based on Schwarz symmetrization. Theorem [Aubin, JDG; Talenti, AMPA; 1976]:  If 1 < p < n andf Cc( ), then ||f ||p  cn,p|| f ||p*

  26. f f µf= µf Schwarz Symmetrization Definition:  The distributionfunctionoffCc( ) isdefinedby µf (t) = V({x : | f(x)| > t}). The Schwarz symmetralfoffisdefinedby f (x) = sup{t > 0: µf (t) > n ||x||}.

  27. Lp Sobolev Inequality Notation np p*:= n – p The isoperimetric inequality is the geometric core of the proof for every 1 < p < n. Theorem [Aubin, JDG; Talenti, AMPA; 1976]:  If 1 < p < n andf Cc( ), then ||f ||p  cn,p|| f ||p* Remarks: The proof is based on Schwarz symmetrization. UsingthePolya–Szegöinequality ||f ||p  ||f||p theproofisreducedto a 1-dimensional problem.

  28. Sharp Affine Lp Sobolev Inequality 1 1 – – n n Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]:  If 1 < p < n undf Cc( ), then ||Du f ||–n  cn,p|| f ||p* an,p du p Sn – 1 Remarks: The affine Lp Sobolev inequality is affine invariant andstronger than the classical Lp Sobolev inequality. The normalization an,p is chosen such that ||Du f||–n an, p = ||f||p . du p Sn – 1

  29. Sharp Affine Lp Sobolev Inequality 1 1 1 – – – n n n Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]:  If 1 < p < n undf Cc( ), then ||Du f ||–n  cn,p|| f ||p* an,p du p Sn – 1 Proof.Based on affineversionofthePólya–Szegöinequality:  If 1 ≤ p < n andf Cc( ), then ||Du f||–n ||Du f||–n (*) du du .  p p Sn – 1 Sn – 1 [Zhang, JDG 1999] & [LYZ, JDG 2002]. For all p 1 (*) was established by [Cianchi, LYZ, Calc. Var. PDE 2010]. Remark: For each p > 1 a new affine isoperimetric inequality is needed in the proof.

  30. Theorem [Petty, 1971]: If K  , then V(K)n – 1V(*K)  V(B)n – 1V(*B) = |u.v|dS(K,v). = h(L,v)dS(K,v). = h(L,v)pdSp(K,v). "=" only for ellipsoids 1 2 Sn – 1 Sn – 1 Sn – 1 Petty's Projection Inequality Revisited Definition [LYZ, 2000]: Forp>1andK theLpprojectionbodypKisdefinedby o  |u.v| dSp(K,v), p h(pK,u)p = cn, p Sn – 1 LpMinkowski Addition wheretheLpsurfaceareameasureSp(K,.) isdeterminedby Cauchy‘sProjectionFormula: If K  , then h(K +pt.L,.)p = h(K,.)p+ th(L,.)p V(K +p t.L)–V(K) n h(K,u) = voln – 1(K|u) , Vp(K,L) =lim p t t 0+ where the surface area measure S(K,.) is determined by V(K + tL)–V(K) nV1(K,L) =lim t t 0+

  31. = h(L,v)pdSp(K,v). Sn – 1 Theorem [LYZ, J. Diff. Geom. 2000]: V(K)n/p – 1V(pK)  V(B)n/p – 1V(pB) * * "=" only for centered ellipsoids The Lp Petty Projection Inequality Definition [LYZ, 2000]: If K  , then o Forp>1andK theLpprojectionbodypKisdefinedby o  |u.v| dSp(K,v), p h(pK,u)p = cn, p Sn – 1 wheretheLpsurfaceareameasureSp(K,.) isdeterminedby Remarks: V(K +p t.L)–V(K) n The proof is based on Steiner symmetrization: Vp(K,L) =lim p t t 0+ Sv* K * (SvK). p p 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).

  32. – + c1.pP +pc2.pP Lp Minkowski Valuations Definition: We call :  an Lp Minkowski valuation, if o o (K  L) +p (K  L) = K +p L Notation whenever K  L  . denotes the set of convex polytopes containing the origin. o Theorem [Parapatits, 2011+; Ludwig, TAMS 2005]: A map :  is an SL(n) contravariantLpMinkowskivaluationifandonlyiffor all P , o o o P = for some c1, c2  0.

  33. 1 1 2 2 Remark: The (symmetric)Lp projection body pK is – + pK:= .pK +p.pK. Asymmetric Lp Projection Bodies Definition:  Forp>1andK theasymmetricLpprojection bodypKis defined by o  (u.v)dSp(K,v),  p h(pK,u)p = an, p Sn – 1 where (u.v) = max{u.v, 0}.

  34. "=" only if p = p Theorem [Haberl & S., J. Diff. Geom. 2009]: If pB = B, then V(pK)  V(pK)  V(pK) * * *,  "=" only if p = p General Lp Petty Projection Inequalities Theorem [Haberl & S., J. Diff. Geom. 2009]: If pK is the convex body defined by – + pK = c1.pK +pc2.pK, then V(K)n/p – 1V(pK)  V(B)n/p – 1V(pB) * * "=" only for ellipsoids centered at the origin

  35. "=" only if p = p V(K)n/p – 1V(pK)  V(K)n/p – 1V(pK) *, * "=" only if p = p General Lp Petty Projection Inequalities Theorem [Haberl & S., J. Diff. Geom. 2009]: If pK is the convex body defined by – + pK = c1.pK +pc2.pK, then  V(B)n/p "=" only for ellipsoids centered at the origin Theorem [Haberl & S., J. Diff. Geom. 2009]: If pB = B, then V(pK)  V(pK)  V(pK) * * *, 

  36. Asymmetric Affine Lp Sobolev Inequality ||Du f ||–n ||Du f||–n  2  cn,p|| f ||p* + du du p p Sn – 1 Sn – 1 Notation Remarks: + Duf:= max{Duf, 0} 1 1 1 – – n n p The asymmetric affine Lp Sobolev inequality is stronger than the affine Lp Sobolev inequality of LYZ for p > 1. The affine L2 Sobolev inequality of LYZ is equivalent via an affine transformation to the classiscal L2 Sobolev inequality; the asymmetric inequality is not! Theorem [Haberl & S., J. Funct. Anal. 2009]:  If 1 < p < n andf Cc( ), then

  37. An Asymmetric Affine Polya–Szegö Inequality Remark: 1 1 – – n n 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]. Theorem [Haberl, S. & Xiao, Math. Ann. 2011]:  If p 1 andf Cc( ), then ||Du f||–n ||Du f||–n + + du du  p p Sn – 1 Sn – 1

  38. Sharp Affine Gagliardo-Nirenberg Inequalities  – n Theorem [Del Pino & Dolbeault, JMPA 2002]: [Haberl, S. & Xiao, Math. Ann. 2011]:  If 1 < p < n, p < q < p(n – 1)/(n – p) andf Cc( ), then for suitable r(p,q), (n,p,q) > 0, ||Du f||–n + du ||f ||p   dn,p,q|| f ||q || f ||r   – 1 p Sn – 1 Affine (Asymmetric) Log-Sobolev Inequalities Haberl, Xiao, S. (Math. Ann. '11) Remarks: Other Affine AnalyticInequalitiesinclude … These sharp Gagliardo-Nirenberg inequalities interpolate between the Lp Sobolev and the Lp logarithmic Sobolev inequalities (Del Pino & Dolbeault, J. Funct. Anal 2003). Affine Moser-Trudinger and Morrey-Sobolev Inequalities Cianchi, LYZ (Calc. Var. PDE '10) A proof using a mass-transportation approach was given by Cordero-Erausquin, Nazaret, Villani (Adv. Math. 2004)

  39. The Orlicz-Petty Projection Inequality h(K,u)dS(K,u)  Definition [LYZ, 2010]: Suppose that :  [0,) is convex and (0) = 0. ForK theOrliczprojectionbodyKisdefinedby o x. u   dV(K,u) ≤ 1 . h(K,x) = inf > 0:  h(K,u) Sn – 1 Normalized Cone Measure 1 VK () = nV(K)

  40. The Orlicz-Petty Projection Inequality Theorem [LYZ, Adv. Math. 2010]: V(K)– 1V(K)  V(B)– 1V(B) * * "=" only for centered ellipsoids Definition [LYZ, 2010]: If K  , then o Suppose that :  [0,) is convex and (0) = 0. ForK theOrliczprojectionbodyKisdefinedby o x. u   dV(K,u) ≤ 1 . h(K,x) = inf > 0:  h(K,u) Sn – 1 Sv* K * (SvK)   Remark: For (t) = |t|p ((t) = max{0,t}p) the Orlicz PPI becomes the (asymmetric) Lp PPI. However, NO CLASS REDUCTION! An Orlicz BPCI was also established by LYZ (J. Diff. Geom. 2010) and later by Paouris & Pivovarov. The proof is based on Steiner symmetrization:

  41. Open Problem – How strong is the PPI really? Question: Suppose that   MValSO(n) has degree n – 1 and B = B. Is it true that V(K)n – 1V(*K) V(K)n – 1V(*K)   V(B)n ? Obstacle: Theorem [Haberl & S., 2011+]: In general Sv* K *(SvK). If n = 2 and  is even, then this is true! Notation: MValSO(n):= { continuousMinkowskivaluation, whichis translation in- and SO(n) equivariant} Work in progress [Haberl & S., 2011+]: If n 3 and is „generated by a zonoid“, then this is true!

More Related