Minimal volume product of convex bodies with various symmetries - PowerPoint PPT Presentation

Minimal volume product of convex bodies with various symmetries Masataka Shibata Tokyo Institute of Technology 18 September, 2019 Conference on Convex, Discrete and Integral Geometry (Jena) joint work with Hiroshi Iriyeh (Ibaraki Univ.)

Contents 2/15 ◮ Mahler’s conjecture and known results ◮ Generalized problem and main results ◮ Sketch of the proof

Volume product and the Mahler conjecture 3/15 For a convex body K in R n , we denote its polar w.r.t. the origin o by K ◦ . ( K ◦ := { Q ∈ R n ; P · Q ≤ 1 for any P ∈ K } . ) Volume product Let K be a convex body in R n and K ◦ be its polar, P ( K ) := | K || K ◦ | := vol n ( K ) vol n ( K ◦ ) is the volume product of K . Fact: P is invariant with respect to any linear transformation A ∈ GL ( n ) . Mahler’s conjecture [Mahler (1939)] For any centrally symmetric (i.e. K = − K ) convex body K in R n , P ( K ) = | K || K ◦ | ≥ 4 n n ! . Remark ◮ n = 1 case is trivial, n = 2 case was shown by Mahler himself. ◮ Sharp upper estimate of P ( K ) is already known as the Blacshke–Santaló inequality.

Known results 4/15 [Mahler’s conjecture (1939)] P ( K ) ≥ 4 n /n ! for any centrally symmetric convex body K ⊂ R n . [Saint-Raymond, Publ. Math. Univ. Pierre et Marie Curie (1980)] K ⊂ R n : a convex body , K is symmetric w.r.t. all coordinate plane ( ⇔ 1-unconditional). Then P ( K ) ≥ 4 n /n ! . (simple proof: [Meyer, Israel J. Math. (1986)]) [Reisner, Math. Scand. (1985)] K ⊂ R n : a zonoid ⇒ P ( K ) ≥ 4 n /n ! . [Barthe and Fradelizi, Amer. J. Math. (2013)] Sharp lower bounds of P ( K ) under another symmetry. (Details will be described later.) [Artstein-Avidan, Karasev, and Ostrover, Duke Math. J. (2014)] Viterbo’s conjecture (in the context of symplectic geometry) implies Mahler’s conjecture. There are many other related resluts, however, the conjecture is still open.

Known results 5/15 Theorem [Iriyeh and S. (preprint; arXiv:1706.01749v3)] Let K be a centrally symmetric convex body in R 3 . Then, P ( K ) ≥ 4 3 3! = 32 3 , with equality if and only if either K or K ◦ is a parallelepiped. Remark ◮ The keys to prove the theorem are “equipartition” and “signed volume estimate”. ◮ A simplified proof for “equipartition” is given in [Fradelizi, Hubard, Meyer, Roldán-Pensado, and Zvavitch, arXiv:1904.10765]. Motivation ◮ At present, we cannot solve high dimensional cases ( n ≥ 4) , however, “signed volume estimate” is applicable. (Details will be described later.) ◮ To understand deeply about Mahler’s conjecture, we consider a generalized problem.

Problem 6/15 Let G be a discrete subgroup of orthogonal group O ( n ) ⊂ M n ( R ) . K n ( G ) = { K ∈ K n ; gK = K for any g ∈ G } : the set of all G -invariant convex bodies. Problem Find a minimizer of minimizing problem K ∈K n ( G ) P ( K ) . min Remark ◮ Case G = { E, − E } : Mahler’s conjecture. ( G -invariant iff centrally symmetric) �� ± 1 0 �� ... ◮ Case G = : Saint-Raymond’s result. 0 ± 1 Today, we focus the case n = 3 .

Discrete subgroups of O (3) . 7/15 Fact (see, e.g., [Conway–Smith, “On quaternions and Octonoins”] ) Up to conjugacy, discrete subgroups of O (3) are classified as 7 infinite families and 7 polyhedral groups. In Schönflies notation, C ℓ , S 2 ℓ , C ℓh , C ℓv , D ℓ , D ℓd , D ℓh ( ℓ ∈ N ) , T, T d , T h , O, O h , I, I h . � � � � � � � cos ξ − sin ξ 0 1 0 0 1 0 0 ℓ ∈ N , ξ := 2 π � R ℓ := , V := , H := . sin ξ cos ξ − 1 0 0 0 0 1 0 ℓ 0 0 1 0 0 1 0 0 − 1 C ℓ := � R ℓ � , C ℓh := � R ℓ , H � , C ℓv := � R ℓ , V � , S 2 ℓ := � R 2 ℓ H � , D ℓ := � R ℓ , V H � , D ℓd := � R 2 ℓ H, V � , D ℓh := � R ℓ , V, H � . T := { g ∈ SO (3); g △ = △} , T d := { g ∈ O (3); g △ = △} , T h := {± g ; g ∈ T } , O := { g ∈ SO (3); gP 8 = P 8 } , O h := { g ∈ O (3); gP 8 = P 8 } = {± g ; g ∈ O } , I := { g ∈ SO (3); gP 20 = P 20 } , I h := { g ∈ O (3); gP 20 = P 20 } = {± g ; g ∈ I } , △ : a regular tetrahedron (simplex), P 8 : a regular octahedron, P 20 : a regular icosahedron. Remark K is C ℓh -invariant if and only if K is R ℓ -symmetry and H -symmetry.

Known results 8/15 For a set A ⊂ R n , we denote the group of linear isometries of A by O ( A ) := { g ∈ O ( n ); gA = A } . Theorem. [Barthe and Fradelizi, Amer. J. Math. (2013)] (i) Let P be a regular polytope in R n . Then P ( K ) ≥ P ( P ) holds for any O ( P ) -invariant convex body K ⊂ R n ( n ≥ 2) . (ii) Let P i be a regular polytopes or Euclidean balls in R n i with n 1 + · · · + n k = n . Then P ( K ) ≥ P ( P 1 × · · · × P k ) holds for any O ( P 1 ) × · · · × O ( P k ) -invariant convex body K ⊂ R n ( n ≥ 2) . Remark In the paper, they obtained result for equality condision of (i), and they studied also many hyperplane symmetric case. Let ℓ ≥ 3 and P = [ − 1 , 1] × regular ℓ -gon Q . Hence, P is a right prism with regular ℓ -gonal base. Then O ([ − 1 , 1]) × O ( Q ) is D ℓh = � R ℓ , V, H � . By the theorem, P ( K ) ≥ P ( P ) for any D ℓh -invariant convex body K ∈ R 3 . Theorem [Iriyeh-S. in preparation] P ( K ) ≥ P ( P ) for any C ℓh -invariant convex body K ∈ R 3 .

Main result 9/15 Known results G minimizer P D 2 h cube, P 8 [Saint-Raymond (1980)] S 2 cube, P 8 [Iriyeh-Shibata] (Mahler’s conjecture) T d simplex [Barthe-Fradelizi (2013)] O h cube, P 8 [Barthe-Fradelizi (2013)] I h P 12 , P 20 [Barthe-Fradelizi (2013)] D ℓh ( ℓ ≥ 3) regular ℓ -prism, regular ℓ -bipyramid [Barthe-Fradelizi (2013)] Main Theorem [Iriyeh-S, in preparation] P ( K ) ≥ P ( P ) holds for G -invariant convex body K . G minimizer P G minimizer P T simplex C 2 h , T h , S 6 , D 3 d cube, P 8 O cube, P 8 C ℓh , D ℓ ( ℓ ≥ 3) regular ℓ -prism, regular ℓ -bipyramid I P 12 , P 20 Remaining cases and conjecture G minimizer P { id. } , C 1 v , C 2 , C 2 v simplex C ℓ , C ℓv ( ℓ ≥ 3) regular ℓ -pyramid S 2 ℓ , D ℓd ( ℓ = 2 , ℓ ≥ 4) regular ℓ -antiprism, its polar

Key lemma: Signed volume estimate 10/15 In [Iriyeh-S.], we introduced “signed volume estimate”. Key lemma (signed volume estimate) Assume that ◮ K ⊂ R 3 is a convex body, K ◦ is the polar of K . o ∗ S ◮ S ⊂ ∂K with piecewise C 1 boundary C = ∂S . S ◮ S ◦ ⊂ ∂K ◦ with piecewise C 1 boundary C ◦ = ∂S ◦ . Then o | o ∗ S | 3 | o ∗ S ◦ | 3 ≥ 1 3 2 C · C ◦ . Here o ∗ S := { λx ; x ∈ S, 0 ≤ λ ≤ 1 } is the truncated cone over S , and C is a vector valued line integral C := 1 � r × dr , where r is a parametrization of C . o ∗ S ◦ and C ◦ 2 C are determined similarly. Remark If C is a curve (not necessary closed) on a plane H with o ∈ H , then C is a normal vector of H and | C | = | o ∗ C | 2 .

Proof of Key lemma 11/15 Using smooth approximation, we can assume K is a smooth strongly convex body. Put Λ( x ) := ∇ µ K ( x ) , where µ K is the Minkowski gauge. Then Λ : ∂K → ∂K ◦ : smooth diffeomorphism. Moreover, x · Λ( x ) = 1 for any x ∈ ∂K . dx = 1 � � | o ∗ S | 3 = x · n ( x ) dS ( x ) (the divergence theorem, div x = 3 ) 3 o ∗ S ∂ ( o ∗ S ) = 1 � x · n ( x ) dS ( x ) ( x · n ( x ) = 0 on ∂ ( o ∗ S ) \ S ) 3 S = 1 1 � | Λ( x ) | dS ( x ) ( n ( x ) = Λ( x ) / | Λ( x ) | ) 3 S where n ( x ) is the unit normal vector at x . Thus, we have 1 1 � � 3 2 | o ∗ S | 3 | o ∗ S ◦ | 3 = | Λ − 1 ( x ◦ ) | dS ( x ◦ ) | Λ( x ) | dS ( x ) S S ◦ Λ − 1 ( x ◦ ) Λ( x ) � � | Λ − 1 ( x ◦ ) | dS ( x ◦ ) ≥ | Λ( x ) | dS ( x ) · S S ◦ (Λ( x ) ∈ K ◦ , Λ − 1 ( x ◦ ) ∈ K, Λ( x ) · Λ − 1 ( x ◦ ) ≤ 1) � � S ◦ n ( x ◦ ) dS ( x ◦ ) = 1 � � ∂S ◦ r ◦ × dr ◦ . = n ( x ) dS ( x ) · r × dr · 4 S ∂S (the Stokes theorem) �

Sketch of the proof: Case G = C ℓh = � R ℓ , H � ( ℓ ≥ 3) : Setting 12/15 Using smooth approximation, we can assume K is a G -invariant smooth strongly convex body. We use same Λ in the proof of Key Lemma. (Put Λ( x ) := ∇ µ K ( x ) , where µ K is the Minkowski gauge. Then Λ : ∂K → ∂K ◦ : smooth diffeomorphism. Moreover, x · Λ( x ) = 1 for any x ∈ ∂K .) We denote by cone( A 1 , . . . , A k ) the polyhedral cone generated by A 1 , . . . , A k . ( cone( A 1 , . . . , A k ) := { λ 1 A 1 + · · · + λ k A k ; λ 1 , . . . , λ k ≥ 0 } .) Up to linear transformation, we can assume � cos 2 π/ℓ � 0 � � 1 � � P := , A := , B := sin 2 π/ℓ ∈ ∂K 0 0 1 0 0 ˆ K := K ∩ cone( P, A, B ) , S := ∂K ∩ cone( P, A, B ) , S ◦ := Λ( S ) ⊂ ∂K ◦ , K ◦ := o ∗ S ◦ . ˆ C ( P, A ) := conv( P, A ) ∩ ∂K : a curve on ∂K , from P to A . C ( A, B ) := conv( A, B ) ∩ ∂K : a curve on ∂K , from A to B . C ( B, P ) := conv( B, P ) ∩ ∂K : a curve on ∂K , from B to P .