Brunn-Minkowski type inequalities and conjectures K aroly B or - PowerPoint PPT Presentation

Brunn-Minkowski type inequalities and conjectures K´ aroly B¨ or¨ oczky Alfr´ ed R´ enyi Institute of Mathematics and CEU Jena, September, 2019

Brunn-Minkowski inequality K , C convex bodies in R n , α, β > 0 α K + β C = { α x + β y : x ∈ K , y ∈ C } { x ∈ R n : � u , x � ≤ α h K ( u ) + β h C ( u ) ∀ u ∈ S n − 1 } = Brunn-Minkowski inequality α, β > 0 1 1 1 n ≥ α V ( K ) n + β V ( C ) V ( α K + β C ) n with equality iff K and C are homothetic ( K = γ C + x , γ > 0). Equivalent form λ ∈ (0 , 1) V ((1 − λ ) K + λ C ) ≥ V ( K ) 1 − λ V ( C ) λ .

Optimal transportation to prove B-M inequality V ( K ) = V ( C ) = 1, K , C convex bodies in R n Caffarelli, Brenier ∃ C ∞ convex ϕ : int K → R such that T = ∇ ϕ : int K → int C bijective & det ∇ T = det ∇ 2 ϕ = 1 Gromov’s argument for Brunn-Minkowski (appendix to Milman-Schechtman) λ ∈ (0 , 1), y = (1 − λ ) x + λ T ( x ) ∈ (1 − λ ) K + λ C = ⇒ dy = det[(1 − λ ) I n + λ ∇ T ( x )] dx � � V ((1 − λ ) K + λ C ) ≥ det[(1 − λ ) I n + λ ∇ T ( x )] dx ≥ 1 dx = 1 K K det[(1 − λ ) A + λ B ] ≥ (det A ) 1 − λ (det B ) λ for positive definite A , B Figalli, Maggi, Pratelli - stability of Brunn-Minkowski (strongest version by Kolesnikov, Milman)

Surface area measure, Minkowski’s first inequality S K - surface area measure on S n − 1 of a convex body K in R n ◮ ∂ K is C 2 + = ⇒ dS K = κ − 1 d H n − 1 κ ( u ) =Gaussian curvature at x ∈ ∂ K where u is normal. ◮ K polytope, F 1 , . . . , F k facets, u i exterior unit normal at F i S K ( { u i } ) = H n − 1 ( F i ) . Minkowski’s first inequality If V ( K ) = V ( C ), then � � S n − 1 h C dS K ≥ S n − 1 h K dS K , with equality iff K and C are translates.

Minkowski problem - characterize S K Given Borel measure µ on S n − 1 with � S n − 1 u d µ ( u ) = o =origin, to solve the Minkowski problem finding K with µ = S K , ◮ Minimize � S n − 1 h C d µ under the condition V ( C ) = 1 ◮ Uniqueness up to translation comes from uniqueness in the Minkowski inequality Monge-Ampere type differential equation on S n − 1 : det( ∇ 2 h + h I n − 1 ) = κ − 1 where h ( u ) = h K ( u ) = max {� u , x � : x ∈ K } support function. Curvature function For any convex body K , f K ( u ) = det( ∇ 2 h K ( u ) + h K ( u ) I n − 1 ) for H n − 1 a.e. u ∈ S n − 1

Decomposition of Surface area measure Lebesgue’s decomposition of S K for a convex body K S K = S a K + S s K where S s K singular K = f K d H n − 1 dS a Minkowski problem for curvature functions Given positive continuous f on S n − 1 � f = f K for a convex body K ⇐ ⇒ S n − 1 u · f ( u ) du = o Regularity theory of Monge-Ampere Given dS K = f K d H n − 1 , f K > 0 ◮ f K is C α for α ∈ (0 , 1] ⇐ ⇒ ∂ K is C 2 ,α + ◮ f K is C k for k ≥ 1 ⇐ ⇒ ∂ K is C k +2 +

?B-M type inequality for affine surface area? Monika Ludwig, Thomas Wannerer, Andrea Colesanti, K.B. Affine surface area � n � d H n − 1 = 1 n +1 d H n − 1 ( x ) n +1 Ω( K ) = S n − 1 f κ ( x ) K ∂ K Theorem (Lutwak) If n = 2 and α, β > 0, then 3 3 3 2 ≥ α Ω( K ) 2 + β Ω( C ) 2 , Ω( α K + β C ) with equality if and only if K and C are homothetic. (Counter)example For n ≥ 3, there exist o -symmetric K and C n +1 n +1 n +1 n ( n − 1) < Ω( K ) n ( n − 1) + Ω( C ) n ( n − 1) . Ω( K + C )

Curvature image bodies Any convex body M in R n has a unique Santalo point s ( M ) ∈ int M such that z ∈ int M V (( M − z ) ∗ ) = V (( M − s ( M )) ∗ ) . min = ⇒ � S n − 1 u · h M − s ( M ) ( u ) − ( n +1) d H n − 1 ( u ) = o . Minkowski problem = ⇒ ∃ convex body CM (curvature image) f CM ( u ) = h M − s ( M ) ( u ) − ( n +1) for u ∈ S n − 1 . Theorem (Lutwak, Schneider) If K , M convex bodies and K ⊂ CM , then Ω( K ) ≤ Ω( CM ) , with equality if and only if K = CM .

Affine surface area and curvature image bodies Monika Ludwig, Thomas Wannerer, Andrea Colesanti ∂ M is C 2 ⇒ ∂ ( CM ) is C 4 + = + (Monge-Ampere equations). Theorem α, β > 0 and N = CM for a convex body M with C 2 + boundary. There exists δ > 0 such that if the C 4 distance of convex bodies K and C with C 4 boundary is less than δ from N , then n +1 n +1 n +1 n ( n − 1) ≥ α Ω( K ) n ( n − 1) + β Ω( C ) n ( n − 1) , Ω( α K + β C ) with equality if and only if K and C are homothetic.

?B-M type inequality for p -affine surface area? Monika Ludwig, Thomas Wannerer, Andrea Colesanti p -Affine surface area p � = − n and o ∈ int K (Hug, Ludwig) n (1 − p ) � n � − p d H n − 1 = S n − 1 ( h n +1 n + p dV K n + p n + p Ω p ( K ) = f K ) S n − 1 h f K K K Theorem n = 2, 2 3 ≤ p ≤ 1, α, β > 0, o ∈ int K , o ∈ int C 2+ p 2+ p 2+ p 2(2 − p ) ≥ α Ω p ( K ) 2(2 − p ) + β Ω p ( C ) 2(2 − p ) . Ω p ( α K + β L ) If 2 3 ≤ p < 1, then equality holds if and only if K and C are dilates. Remark Seems to fail completely if p < 2 3 or p > 1

Logarithmic Minkowski problem - Cone volume measure n h K dS K - cone volume measure on S n − 1 if o ∈ K dV K = 1 (Gromov, Milman, 1986) - also called L 0 surface area measure ◮ K polytope, F 1 , . . . , F k facets, u i exterior unit normal at F i V K ( { u i } ) = h K ( u i ) H n − 1 ( F i ) = V ( conv { o , F i } ) . n ◮ V K ( S n − 1 ) = V ( K ). Monge-Ampere type differential equation on S n − 1 for h = h K if µ has a density function f : h det( ∇ 2 h + h I ) = f B. Lutwak, Yang, Zhang solved in the even case

Logarithmic ( L 0 ) Brunn-Minkowski conjecture λ ∈ [0 , 1], o ∈ int K , int C (1 − λ ) K + 0 λ C = { x ∈ R n : � u , x � ≤ h K ( u ) 1 − λ h C ( u ) λ ∀ u ∈ S n − 1 } λ K + 0 (1 − λ ) C ⊂ λ K + (1 − λ ) C Conjecture (Logarithmic Brunn-Minkowski conjecture) λ ∈ (0 , 1) , K, C are o-symmetric V ((1 − λ ) K + 0 λ C ) ≥ V ( K ) 1 − λ V ( C ) λ with equality iff K and C have dilated direct summands. Conjecture (Logarithmic Minkowski conjecture) For o-symmetric K, C, if V ( K ) = V ( C ) , then � � S n − 1 log h C dV K ≥ S n − 1 log h K dV K , with equality iff K and C have dilated direct summands.

Known cases of the logarithmic B-M conjecture 1 ◮ Interesting for any log-concave measure (like Gaussian) instead of volume log B-M conjecture for volume = ⇒ log B-M conjecture for any log-concave measure (Saroglou) ◮ n = 2 for volume (Stancu + BLYZ) ◮ K and C are unconditional for any log-concave measure - follows directly from Pr´ ekopa-Leindler (Bollob´ as&Leader + Cordero-Erausquin&Fradelizi&Maurey + Saroglou on coordinatewise product) ◮ K and C are dilates for the Gaussian measure (Cordero-Erausquin&Fradelizi&Maurey on B -conjecture) ◮ Holds for the volume in R 2 n = C n if K and C are complex convex bodies (Rotem)

Logarithmic B-M conjecture for almost ellipsoids Chen, Huang, Li, Liu verified logarithmic B-M conjecture based on a result by Milman-Kolesnikov if K is close to be an ellipsoid: ∃ ε n > 0 such that if K , C o -symmetric with V ( K ) = V ( C ) and E ⊂ K ⊂ (1 + ε n ) E for an ellipsoid E , then � � S n − 1 log h C dV K ≥ S n − 1 log h K dV K , with equality iff C = K .

Consequences of the log-B-M conjecture - Gardner-Zvavitch Conjecture Livshyts, Marsiglietti, Nayar, Zvavitch logarithmic B-M conjecture = ⇒ Gardner-Zvavitch Conjecture 1 1 1 n ≥ αγ ( K ) n + (1 − α ) γ ( C ) γ ( α K + (1 − α ) C ) n for o -symmetric K , C and the Gaussian measure γ on R n . ( γ can be replaced by any even log-concave measure) Theorem (Kolesnikov, Livshyts) � � K x d γ ( x ) = o and C x d γ ( x ) = o = ⇒ 1 1 1 2 n ≥ αγ ( K ) 2 n + (1 − α ) γ ( C ) γ ( α K + (1 − α ) C ) 2 n

L p surface area measures L p surface area measures (Lutwak 1990) p ∈ R dS K , p = h 1 − p dS K = nh − p K dV K K Examples ◮ S K , 1 = S K ◮ S K , 0 = nV K ◮ S K , − n related to SL ( n ) invariant f K ( u ) h K ( u ) n +1 Theorem (Chou&Wang,Chen&Li&Zhu,B&Bianchi&Colesanti) If p > 0 , p � = 1 , n, then any finite Borel measure µ on S n − 1 not concentrated on any closed hemisphere is of the form µ = S K , p . Remark S n − 1 h p ◮ Minimize � C d µ under the condition V ( C ) = 1 ◮ Conjectured to be unique in the even case if 0 < p < 1

L p Brunn-Minkowski inequality/conjecture p > 0, λ ∈ (0 , 1), o ∈ int K , int C λ K + p (1 − λ ) C = { x ∈ R n : � u , x � p ≤ λ h K ( u ) p +(1 − λ ) h C ( u ) p ∀ u } � 1 / p λ h p K + (1 − λ ) h p � p ≥ 1 h λ K + p (1 − λ ) C = C L p Brunn-Minkowski inequality/conjecture p p p n ≥ λ V ( K ) n + (1 − λ ) V ( C ) V ( λ K + p (1 − λ ) C ) n with equality iff K and C are dilated. Equivalent V ( λ K + p (1 − λ ) C ) ≥ V ( K ) λ V ( C ) 1 − λ Theorem ( p > 1, Firey, 1962) L p Brunn-Minkowski inequality holds if o ∈ int K , int C Conjecture (0 < p < 1, BLYZ, 2012) L p Brunn-Minkowski inequality holds if K and C are o-symmetric. L 0 = ⇒ L p for 0 < p < 1, L 1 = ⇒ L p for p > 1