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 If K following , thenfunctional on nn – 1 S(K ) n K 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 ) du. | S(K ) = vol (K u 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 polar body K of K* is defined by ( h( *K, u) == volnn–– 11(K (K||uu ))– 1 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: If K , then The BPCI of the + 1) – (n Random-Simplex + 1) V(Kis) –a(nreformulation V( K ) V(B) 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 + t 1 L) t. L) – V(K ) ) n. V– 11(K – n. V (K, , LL)) == 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 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 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: If Kt , K 1 t , …, Knt 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: If Kt , K 1 t , …, Knt 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: If Kt , K 1 t , …, Knt 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: If Kt , K 1 t , …, Knt 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: Kt = [– x, x] 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 valuation if map : : is called a Minkowski if L) L) ++ (K (K L) L) = = (K (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 – n 1 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 – n 1 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 – n 1 cn, p || f || p* Proof. Based on affine version of the Pólya – Szegö inequality: inequality If 1 ≤ p < n and f Cc ( ), then (*) –n || D f || u p du Sn – 1 – n 1 –n || D f || u p du Sn – 1 [Zhang, JDG 1999] & [LYZ, JDG 2002]. p p 1>(1 by Remark: For all each newestablished aff ine isoperimetric *)awas [Cianchi, Calc. in Var. 2010]. inequality. LYZ, is needed the. PDE proof. – n 1 .
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]: ) = 1. v| d. S(K, v). , | |u h( K, u) = vol (K u n – 1 2 S body p K is def ined by For p > 1 and K o the Lp projection n– 1 where the surface area p measure S(K, . . ) isp determined by h( p K, u) = cn, p | u v | d. S (K, v), p Sn – 1 V(K + t L) – V(K ) n. Vthe , L )surface = . ) ish(L, v) d. S(K, v). = lim area measure S (K, 1(KL where t 0 n – 1 determined by t p p S + 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]: is basedthe on L Steiner symmetrization: For. The p > proof 1 and K projection body K is def ined by o p *p K *p (S K ). Sv v 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 the by Lp surface measure p(K, ) is determined by Lp PPI LYZ (J. area Diff. Geom. S 2000). t. L) – V(K ) An direct proof of. V(K the+L BPCI using Shadow p. Systems pp V (K , L ) = lim = h(L, v) d. Sp(K, v). p pgiven by t 0 – 1 was Campi & Gronchi (Adv. 2002). t S n. Math. +
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, Notation. TAMS 2005]: A map : o o is an SL(n) contravariant Lpset Minkowski denotes the of convex o containing the origin. valuation if and only if for all P polytopes 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 L projection body K p p o h( p K, u) p = an, p S . v ) p d. Sp(K, v), ( u 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, – 1 n/p *, V(K )n/p – 1 V( p* K V(K ) )n/p V(B) 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 – n 1 ), then –n –n + 2 || D f || u p du Sn – 1 Remarks: Sn – 1 – n 1 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 –n + || D f || u p du Sn – 1 – n 1 –n + || D f || u p du – n 1 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. &&Dolbeault, JMPA [Haberl, Xiao, Math. Ann. 2002]: 2011]: 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 – 1 + –n || f || d || f || r || D f || p, q q u p p n, du Sn – 1 Other Remarks: Affine Analytic Inequalities include … These Gagliardo-Nirenberg inequalities interpolate Affine sharp (Asymmetric) Log-Sobolev Inequalities between the Lp. S. Sobolev Lp logarithmic Sobolev Haberl, Xiao, (Math. and Ann. the '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. Nazaret, Var. PDE '10) (Adv. Math. 2004) Cordero-Erausquin, Villani
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 p ( (t) p) the Orlicz PPI For (t) = NO |BPCI = also max{0, t}symmetrization: Remark: The However, CLASS REDUCTION! proof ist | based on Steiner An Orlicz was established by LYZ * K and becomes the PPI. (J. Diff. Geom. by S (asymmetric) 2010) * (later SLp K ) Paouris & Pivovarov. 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 *Kn) ? V(K *)n. K–)1 V( V(K )n – 1 V( V(B) Theorem [Haberl & S. , 2011+]: If n = 2 and is even, then this is true! Obstacle: Notation: SO(n) : In general MVal = { continuous Minkowski valuation, which is Work in progress [Haberl & S. , 2011+]: * K in*(Sand SO(n) equivariant} Stranslation equivariant v v K ). If n 3 and is „generated by a zonoid“, then this is true!
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