The Petty Projection Inequality and BEYOND Franz Schuster

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

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

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

Polar Projection Bodies – The PPI Reformulated Def inition [Minkowski, 1900]: The projection body K of K is defined by h( K, u) = voln – 1(K | u ) L is a zonoid if L = K + t for some K Zonoids in … Support Function h(K, u) = max{u. x: x K} , t .

Polar Projection Bodies – The PPI Reformulated Def inition [Minkowski, 1900]: * The projection body K of K is defined by polar ( *K, u) = volnn–– 11(K ||u ))– 1 h( K, u) = vol (K u Radial functions Polar projection bodies (K, u) = max{ 0: u K} Theorem [Petty, 1971]: If K *K : = ( K )* , then V(K ) n – 1 V( *K ) V(B) n – 1 V( *B) "=" only for ellipsoids

The Busemann-Petty Centroid Inequality – Class Reduction Def inition [Dupin, 1850]: The centroid body K of K is defined by h( K, u) = | x. u | dx. K Theorem [Petty, Pacific J. Math. 1961]: Remarks: then If K , The BPCI is–a(nreformulation of the Random-Simplex V(K ) + 1)V( K ) V(B) – (n + 1)V( B) Inequality by Busemann (Pacific J. Math. 1953). "=" only for centered ellipsoids Petty deduced the PPI from the BPCI!

The Busemann-Petty Centroid Inequality – Class Reduction [Lutwak, Trans. AMS 1985]: BPCI for polars of zonoids PPI for all convex bodies PPI for zonoids BPCI for all star bodies Based on V 1( K , L ) = 2 n+1 V– 1(L, *K ), where V(K 1 t – – V(K + t L). L)V(K ) ) n. V 1(K L ) = lim – n. V– 1(K , , L ) = lim t t 00 tt ++ Harmonic Radial Addition (K 1 t. L, . ) – 1 = (K, . ) – 1 + t (L, . ) – 1

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Def inition [Rogers & Shephard, 1958]: 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}, v A t [0, 1].

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Def inition [Rogers & Shephard, 1958]: 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}, v t [0, 1].

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

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Properties of Shadow Systems: t t If Kt , K 1 , …, Kn are shadow systems, then V(K 1 t, …, Knt ) is convex in t, in particular V(Kt) is convex Steiner symmetrization is a special volume preserving shadow system Mixed Volumes V( 1 K 1 + … + m. Km) = … i 1 in V(Ki 1, …, Kin )

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Properties of Shadow Systems: t t If Kt , K 1 , …, Kn are shadow systems, then V(K 1 t, …, Knt ) is convex in t, in particular V(Kt) is convex Steiner symmetrization is a special volume preserving shadow system K v v

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Properties of Shadow Systems: t t If Kt , K 1 , …, Kn are shadow systems, then V(K 1 t, …, Knt ) is convex in t, in particular V(Kt) is convex Steiner symmetrization is a special volume preserving shadow system K Sv K = K v v 1 2

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 Properties of Shadow Systems: t t If Kt , K 1 , …, Kn are shadow systems, then V(K 1 t, …, Knt ) is convex in t, in particular V(Kt) is convex Steiner symmetrization is a special volume preserving shadow system K Sv K = K v v 1 2 K 1

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 First step: K Kt = [– x, x] dx dx Kt implies V( K ) = … V([– x 1, x 1], …, [– xn , xn]) dx 1…dxn. K K

A Proof of the BPCI – Campi & Gronchi, Adv. Math. 2002 First step: Kt = [– x, x] dx Kt implies V( Kt ) = … V([– x 1, x 1]t , …, [– xn , xn]t ) dx 1…dxn. K K Second step: V( (Sv K )) = V( K ) 1 2 V( K 0) + 1 2 V( K 1) Since V( K 0) = V( K ) and V( K 1) = V( K ) this yields V( (Sv K )) V( K ).

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

Valuations on Convex Bodies Def inition: A function : is called a Minkowski valuation if is called a valuation if map : (K L) + (K L) = (K ) + (L) whenever K L . Theory of Valuations: Abardia, Alesker, Bernig, Fu, Goodey, Groemer, Haberl, Hadwiger, Hug, Ludwig, Klain, Mc. Mullen, Parapatits, Reitzner, Schneider, Wannerer, Weil, …

Valuations on Convex Bodies Def inition: A map : is called a Minkowski valuation if (K L) + (K L) = (K ) + (L) whenever K L . Examples: Trivial examples are Id and – Id is a Minkowski valuation

Classif ication of Minkowski Valuations Theorem [Haberl, J. EMS 2011]: A map : o is a continuous and SL(n) contravariant Minkowski valuation if and only if = c for some c 0. Remarks: SL(n) contravariance (AK ) = A – T (K ), A SL(n) The map : o is the only non-trivial continuous SL(n) covariant Minkowski valuation. First such characterization results of and were obtained by Ludwig (Adv. Math. 2002; Trans. AMS 2005).

The Isoperimetric and the Sobolev Inequality: If f Cc ( ), then 1 n || f ||1 n n || f || n n– 1 [Federer & Fleming, Ann. Math. 1960] [Maz‘ya, Dokl. Akad. Nauk SSSR 1960] Notation || f || p = Isoperimetric Inequality: 1 n S(K ) n n V(K ) n– 1 n | f (x)| p dx 1/p

Aff ine Zhang – Sobolev Inequality Theorem [Zhang, J. Diff. Geom. 1999]: If f Cc ( || f ||1 ), then n n 2 n – 1 S –n || Du f ||1 du n– 1 1 –n 1 n n n || f || n n– 1 Notation Remarks: Du f : = u. f The aff ine Zhang –Sobolev inequality is aff ine invariant and equivalent to an extended Petty projection inequality. It is stronger than the classical Sobolev inequality.

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

Schwarz Symmetrization Def inition: The distribution function of f C ( c µf (t) = V({x ) is def ined by : | f (x)| > t}). The Schwarz symmetral f of f is def ined by f (x) = sup{t > 0: µf (t) > n ||x||}. f f µf = µ f

Lp Sobolev Inequality Theorem [Aubin, JDG; Talenti, AMPA; 1976]: If 1 < p < n and f Cc ( ), then || f || p cn, p || f || p* Notation np *: p = n–p Remarks: The proof is based on Schwarz symmetrization. Using the Polya – Szegö inequality || f || p the proof is reduced to a 1 -dimensional problem. The isoperimetric inequality is the geometric core of the proof for every 1 < p < n.

Sharp Aff ine Lp Sobolev Inequality Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]: If 1 < p < n und f Cc ( an, p S ), then – || Du f ||p n du 1 –n n– 1 cn, p || f || p* Remarks: The aff ine Lp Sobolev inequality is aff ine invariant and stronger than the classical Lp Sobolev inequality. The normalization an, p is chosen such that an, p S – – || Du f || p n du n– 1 1 n = || f || p.

Sharp Aff ine Lp Sobolev Inequality Theorem [Lutwak, Yang, Zhang, J. Diff. Geom. 2002]: If 1 < p < n und f Cc ( an, p S ), then – || Du f ||p n du n– 1 1 –n cn, p || f || p* Proof. Based on affine version of the Pólya – Szegö inequality: inequality If 1 ≤ p < n and f Cc ( ), then (*) || Du f ||p– n du Sn – 1 1 –n – || Du f || p n du Sn – 1 [Zhang, JDG 1999] & [LYZ, JDG 2002]. Remark: For all p 1 (1 )awas established by each p > * new aff ine isoperimetric [Cianchi, LYZ, Calc. in the proof. inequality is needed Var. PDE 2010]. 1 –n .

Petty's Projection Inequality Revisited Theorem [Petty, 1971]: If K , then V(K ) n – 1 V( *K ) V(B) n – 1 V( *B) "=" only for ellipsoids Lp Minkowski Addition Cauchy‘s Projection Formula: If K p, then p p h(K +p t. L, . ) = h(K, . ) + t h(L, . ) Def inition [LYZ, 2000]: h( K, u) volthe 1(K |projection body K is def ined by = u ) = 1 |u. v| d. S(K, v). , n– L 2 S For p > 1 and K p p o n– 1 where the surface area = c p measure S(K, . . ) is determined by h( p K, u) | u v | pd. Sp(K, v), n, p Sn – 1 V(K + t L) – V(K ) n. V 1(K , surface = h(L, v) d. S(K, v). where the LL ) = tlim area measure Sp(K, . S) –is determined by 0 n 1 t p + V(K n V (K , L ) = lim+ p p t 0 +p t. L) – V(K ) = h(L, v) p d. Sp(K, v). t Sn – 1

The Lp Petty Projection Inequality Theorem [LYZ, J. Diff. Geom. 2000]: If K o , then * * V(K )n/p – 1 V( p K ) V(B)n/p – 1 V( p B) "=" only for centered ellipsoids Remarks: Def inition [LYZ, 2000]: on Steiner symmetrization: For. The proof K p > 1 and is basedthe L projection body K is def ined by o p p p Sv * K * (Sv K ). p p p h( p K, u) = cn, p | u. v | d. Sp(K, v), Sn – 1 Via Class Reduction an Lp BPCI was deduced from the where. PPI by surface area measure Sp(K, . ) is determined by Lp the Lp LYZ (J. Diff. Geom. 2000). V(K +p t BPCI using Shadow p. Systems A direct proof of the Lp. L) – V(K ) n V (K , L ) = lim = h(L, v) d. Sp(K, v). p given byt 0 – 1 was p Campi & Gronchi (Adv. S n. Math. 2002). t +

Lp Minkowski Valuations Def inition: We call : o o an Lp Minkowski valuation, if (K L) +p (K L) = K +p L whenever K L . Theorem [Parapatits, 2011+; Ludwig, TAMS 2005]: Notation A map : o o is an SL(n) contravariant Lpset of convex denotes the Minkowski o valuation if and only if for all P polytopes containing the origin. o , + P = c 1. p P +p c 2. p–P for some c 1, c 2 0.

Asymmetric Lp Projection Bodies Def inition: For p > 1 and K is def ined by the asymmetric Lp projection body p K o h( p K, u) p = an, p S p ( u. v ) d. Sp(K, v), n– 1 where (u. v) = max{ u. v, 0}. Remark: The (symmetric) Lp projection body p K is – + p K : = 1. p K + p 1. p K. 2 2

General Lp Petty Projection Inequalities Theorem [Haberl & S. , J. Diff. Geom. 2009]: If p K is the convex body def ined by then – + p K = c 1. p K +p c 2. p K, * * V(K )n/p – 1 V( p K ) V(B)n/p – 1 V( p B) "=" only for ellipsoids centered at the origin Theorem [Haberl & S. , J. Diff. Geom. 2009]: If p B = B, then * * * V( p K ) V( p , K ) "=" only if p = p

General Lp Petty Projection Inequalities Theorem [Haberl & S. , J. Diff. Geom. 2009]: If p K is the convex body def ined by then – + p K = c 1. p K +p c 2. p K, * V(K V(B)n/p * V(K )n/p – 1 V( p K ) )n/p – 1 V( p, K ) "=" only for ellipsoids centered at the origin Theorem [Haberl & S. , J. Diff. Geom. 2009]: If p B = B, then * * * V( p K ) V( p , K ) "=" only if p = p

Asymmetric Aff ine Lp Sobolev Inequality Theorem [Haberl & S. , J. Funct. Anal. 2009]: If 1 < p < n and f Cc ( 1 p 1 –n ), then 2 + || Du f || – n du || Du f ||p– n du p Sn – 1 Remarks: Sn – 1 1 –n cn, p || f || p* Notation + Du f : = max{Du f , 0} The asymmetric aff ine Lp Sobolev inequality is stronger than the aff ine Lp 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!

An Asymmetric Aff ine Polya –Szegö Inequality Theorem [Haberl, S. & Xiao, Math. Ann. 2011]: If p 1 and f Cc ( ), then + || Du f ||p– n du Sn – 1 1 –n – + || Du f || p n du 1 –n Sn – 1 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].

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

The Orlicz-Petty Projection Inequality Def inition [LYZ, 2010]: Suppose that : [0, ) is convex and (0) = 0. For K o the Orlicz projection body K is def ined by h( K, x) = inf x. u > 0: n– 1 h(K, u) S d. V(K, u) ≤ 1. Normalized Cone Measure 1 VK ( ) = n. V(K ) h(K, u) d. S(K, u)

The Orlicz-Petty Projection Inequality Def inition [LYZ, 2010]: Suppose that : [0, ) is convex and (0) = 0. For K o the Orlicz projection body K is def ined by h( K, x) = inf x. u > 0: n– 1 h(K, u) S d. V(K, u) ≤ 1. Theorem [LYZ, Adv. Math. 2010]: If K o , then * * V(K )– 1 V( K ) V(B)– 1 V( B) "=" only for centered ellipsoids For proof t p CLASS REDUCTION! Remark: The. Orlicz NO ( (t) onalso established Orlicz PPI However, |BPCI was max{0, t} p) the by LYZ An (t) = is| based = Steiner symmetrization: becomes. Geom. 2010) and (later by Paouris & Pivovarov. (J. Diff. the (asymmetric) S p K ) S * K * L PPI. v v

Open Problem – How strong is the PPI really? Question: Suppose that MVal. SO(n) has degree n – 1 and B = B. Is it true that V(K ) K V(B) ) V(K )n – 1 V( * n –)1 V( *Kn ? Theorem [Haberl & S. , 2011+]: If n = 2 and Obstacle: is even, then this is true! Notation: In general) : = { continuous MVal. SO(nprogress [Haberl. Minkowski valuation, which is Work in & S. , 2011+]: * K *(Sand). translation in- v K SO(n) equivariant} Sv equivariant If n 3 and is „generated by a zonoid“, then this is true!