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625 Discrete Geometry and Algebraic Combinatorics AMS Special Session Discrete Geometry and Algebraic Combinatorics January 11, 2013 San Diego, CA Alexander Barg Oleg R Musin Editors American Mathematical Society Discrete Geometry and Algebraic Combinatorics AMS Special Session Discrete Geometry and Algebraic Combinatorics January 11, 2013 San Diego, CA Alexander Barg Oleg R Musin Editors www.TechnicalBooksPDF.com www.TechnicalBooksPDF.com 625 Discrete Geometry and Algebraic Combinatorics AMS Special Session Discrete Geometry and Algebraic Combinatorics January 11, 2013 San Diego, CA Alexander Barg Oleg R Musin Editors American Mathematical Society Providence, Rhode Island www.TechnicalBooksPDF.com EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash C Misra Martin J Strauss 2010 Mathematics Subject Classification Primary 52C35, 52C17, 05B40, 52C10, 05C10, 37F20, 94B40, 58E17 Library of Congress Cataloging-in-Publication Data Discrete geometry and algebraic combinatorics / Alexander Barg, Oleg R Musin, editors pages cm – (Contemporary mathematics ; volume 625) “AMS Special Session on Discrete Geometry and Algebraic Combinatorics, January 11, 2013.” Includes bibliographical references ISBN 978-1-4704-0905-0 (alk paper) Discrete geometry–Congresses Combinatorial analysis–Congresses I Barg, Alexander, 1960- editor of compilation II Musin, O R (Oleg Rustamovich) editor of compilation QA640.7.D575 2014 516 116–dc23 2014007424 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/625 Copying and reprinting Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA Requests can also be made by e-mail to reprint-permission@ams.org Excluded from these provisions is material in articles for which the author holds copyright In such cases, requests for permission to use or reprint should be addressed directly to the author(s) (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2014 by the American Mathematical Society All rights reserved The American Mathematical Society retains all rights except those granted to the United States Government Copyright of individual articles may revert to the public domain 28 years after publication Contact the AMS for copyright status of individual articles Printed in the United States of America ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability Visit the AMS home page at http://www.ams.org/ 10 19 18 17 16 15 14 www.TechnicalBooksPDF.com Contents Preface vii Plank theorems via successive inradii K Bezdek Minimal fillings of finite metric spaces: The state of the art A Ivanov and A Tuzhilin Combinatorics and geometry of transportation polytopes: An update J A de Loera and E D Kim 37 A Tree Sperner Lemma A Niedermaier, D Rizzolo, and F E Su 77 Cliques and cycles in distance graphs and graphs of diameters A M Raigorodskii 93 New bounds for equiangular lines A Barg and W.-H Yu 111 Formal duality and generalizations of the Poisson summation formula ă rmann H Cohn, A Kumar, C Reiher, and A Schu 123 On constructions of semi-bent functions from bent functions G Cohen and S Mesnager 141 Some remarks on multiplicity codes S Kopparty 155 Multivariate positive definite functions on spheres O R Musin 177 v www.TechnicalBooksPDF.com www.TechnicalBooksPDF.com Preface This volume contains a collection of papers presented at, or closely related to the topics of, the Special Session on “Discrete Geometry and Algebraic Combinatorics” (January 11, 2013) held as a part of 2013 Joint Mathematics Meetings in San Diego, CA The papers in the volume belong to one of the two related subjects in the session’s title, and can be divided into two groups: distance geometry with applications in combinatorial optimization, and algebraic combinatorics, including applications in coding theory In the first area, the paper by K Bezdek discusses the affine plank conjecture of T Bang Bezdek gives a short survey on the status of this problem and proves some partial results for the successive inradii of the convex bodies involved The underlying geometric structures are successive hyperplane cuts introduced several years ago by J Conway and inductive tilings introduced recently by A Akopyan and R Karasev Transportation polytopes arise in optimization and statistics, and also are of interest for discrete mathematics because permutation matrices, Latin squares, and magic squares appear naturally as lattice points of these polytopes The survey by J.A De Loera and E.D Kim is devoted to combinatorial and geometric properties of transportation polytopes This paper also includes some recent unpublished results on the diameter of graphs of these polytopes and discusses the status of several open questions in this field The paper by A Ivanov and A Tuzhilin presents an overview of a new branch of the one-dimensional geometric optimization problem, the minimal fillings theory This theory is closely related to the generalized Steiner problem and offers an opportunity to look at many classical questions appearing in optimal connection theory from a new point of view The paper is essentially a survey, which serves as a useful introduction to a new theory that so far has been scattered in multiple papers mostly appearing in the Russian literature A.M Raigorodskii presents a survey of recent advances in many classical open problems related to the notion of a geometric graph He discuss some properties ofdistance graphs and graphs of diameters The study of such graphs is motivated by famous problems of combinatorial geometry going back to Erd´ os, Hadwiger, Nelson, and Borsuk The paper by A Niedermaier, D Rizzolo and F.E Su extends the famous Sperner lemma to finite labellings of trees In this paper the authors prove 15 theorems around a tree Sperner lemma In particular they show that any proper labelling of a tree contains a fully-labelled edge and prove that this theorem is equivalent to a theorem for finite covers of metric trees and a fixed point theorem on vii www.TechnicalBooksPDF.com viii PREFACE metric trees They also exhibit connections to Knaster-Kuratowski-Mazurkiewicztype theorems and discuss interesting applications to voting theory In the second area (algebraic combinatorics), A Barg and W.-H Yu use semidefinite programming to obtain new bounds on the maximum cardinality of equiangular line sets in Rn They obtain some new exact answers, resolving in part a 1972 conjecture made by Lemmens and Seidel The Poisson summation formula underlies a number of fundamental results of the theory of codes, lattices, and sphere packings In their paper, H Cohn, A Kumar, C Reiher, and A Schă urmann address the notion of formal duality introduced earlier in the work on energy-minimizing configurations Formal duality is well known in coding theory where several classes of nonlinear codes are formal duals of each other The authors attempt to formalize this notion for the case of packings relying on the Poisson summation formula The paper by G Cohen and S Mesnager is devoted to the classical problem of constructing bent and semi-bent functions This problem has been the focus of attention in computer science in particular because of aplications in cryptography including correlation attacks and linear cryptanalysis The authors construct new families of semi-bent functions and reveal new links between such functions and bent functions In his paper, S Kopparty studies so-called multiplicity codes; i.e., codes obtained by evaluating polynomials at the points of a finite field whereby at each point one computes not just the value of the polynomial but also values of the first few derivatives Such codes were known for about 15 years in the case of univariate polynomials, while recently these ideas were extended to the multivariate case It turns out that these constructions are well suited for local decoding including list decoding procedures O.R Musin presents a new approach to the well-known semidefinite programming bounds on spherical codes Previously these bounds were derived using positive definite matrices, while this paper defines a new class of multivariate orthogonal polynomials that can be used to give a direct proof of the bounds These polynomials satisfy the addition formula as well as positivity conditions generalizing the conditions given the classical Schoenberg theorem for univariate Gegenbauer polynomials A part of the special session was dedicated to the 60th birthday of our friend and colleague Professor Ilya Dumer (UC Riverside) Several authors, including the present editors, also dedicate their papers to Ilya with affection and admiration Alexander Barg University of Maryland Oleg R Musin University of Texas at Brownsville www.TechnicalBooksPDF.com Contemporary Mathematics Volume 625, 2014 http://dx.doi.org/10.1090/conm/625/12489 Plank theorems via successive inradii K´ aroly Bezdek Abstract In the 1930’s, Tarski introduced his plank problem at a time when the field discrete geometry was about to born It is quite remarkable that Tarski’s question and its variants continue to generate interest in the geometric as well as analytic aspects of coverings by planks in the present time as well Besides giving a short survey on the status of the affine plank conjecture of Bang (1950) we prove some new partial results for the successive inradii of the convex bodies involved The underlying geometric structures are successive hyperplane cuts introduced several years ago by Conway and inductive tilings introduced recently by Akopyan and Karasev Introduction As usual, a convex body of the Euclidean space Ed is a compact convex set with non-empty interior Let C ⊂ Ed be a convex body, and let H ⊂ Ed be a hyperplane Then the distance w(C, H) between the two supporting hyperplanes of C parallel to H is called the width of C parallel to H Moreover, the smallest width of C parallel to hyperplanes of Ed is called the minimal width of C and is denoted by w(C) Recall that in the 1930’s, Tarski posed what came to be known as the plank problem A plank P in Ed is the (closed) set of points between two distinct parallel hyperplanes The width w(P) of P is simply the distance between the two boundary hyperplanes of P Tarski conjectured that if a convex body of minimal width w is covered by a collection of planks in Ed , then the sum of the widths of these planks is at least w This conjecture was proved by Bang in his memorable paper [5] (In fact, the proof presented in that paper is a simplification and generalization of the proof published by Bang somewhat earlier in [4].) Thus, we call the following statement Bang’s plank theorem Theorem 1.1 If the convex body C is covered by the planks P1 , P2 , , Pn in n Ed , d ≥ (i.e., C ⊂ P1 ∪ P2 ∪ · · · ∪ Pn ⊂ Ed ), then i=1 w(Pi ) ≥ w(C) In [5], Bang raised the following stronger version of Tarski’s plank problem called the affine plank problem We phrase it via the following definition Let C be a convex body and let P be a plank with boundary hyperplanes parallel to the 2010 Mathematics Subject Classification Primary 52C17, 05B40, 11H31, and 52C45 Partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant c 2014 American Mathematical Society www.TechnicalBooksPDF.com MULTIVARIATE POSITIVE DEFINITE FUNCTIONS ON SPHERES 179 Recall the addition theorem for Gegenbauer polynomials: (n) Gk (cos θ1 cos θ2 + sin θ1 sin θ2 cos ϕ) k (n+2s) = cnks Gk−s (n+2s) (cos θ1 ) Gk−s (cos θ2 ) (sin θ1 )s (sin θ2 )s G(n−1) (cos ϕ), s s=0 where cnks are positive coefficients whose values of no concern here (see [7, 12]) 2.2 Basic definitions Let us denote by u, v the inner product of vectors u, v ∈ Rm , and by |v| we denote the norm of v (i.e |v|2 = v, v ) Definition 2.1 Let ≤ m ≤ n − 2, t ∈ R, u, v ∈ Rm for m > 0, and u = v = for m = Then the following polynomial in 2m + variables of degree k in t is well defined: (n,m) Gk (n−m) (t, u, v) := (1 − |u|2 )k/2 (1 − |v|2 )k/2 Gk (n,0) t − u, v (1 − |u|2 )(1 − |v|2 ) (n,0) (n) Note that by definition we have: Gk (t) = Gk (t, 0, 0) = Gk (t) Let u = (u1 , , un ) ∈ Rn For < m ≤ n we denote by u(m) the vector (u1 , , um ) in Rm Let u(0) = For ≤ m ≤ n − 2, u, v ∈ Rn put (n,m) Gk (n,m) (t, u, v) := Gk (t, u(m) , v(m) ) Remark Note that these multivariate polynomials are different from the multivariate Jacobi polynomials (see [11, 15]) Definition 2.2 We define a matrix Zdm (u, v) of size Rm , v ∈ Rm by Zdm (u, v) := (zdm (u))T zdm (v), where m+d m × m+d m for u ∈ zdm (x) = zdm (x1 , , xm ) := (1, x1 , , xm , x21 , x1 x2 , , xm−1 xm , x2m , , xdm ) is the vector of monomials Denote the inner product of matrices A = aij , B = bij both of size d × d d by A, B , i.e A, B = Tr(AB) = aij bij i,j=1 Definition 2.3 Let f (u, v) be a symmetric polynomial in u, v ∈ Rm , i.e f (u, v) = f (v, u) We say that f (u, v) is positive semidefinite and write f if there are a symmetric matrix H and d ≥ so that f (u, v) = H, Zdm (u, v) Let polynomials fr for all r, fr → f as r → ∞, where a function f (u, v) is symmetric in u, v and continuous on {|u| ≤ 1, |v| ≤ 1} Then we say that f (u, v) is positive semidefinite and write f 2.3 Orthogonality Let t ∈ R, u Denote ⎛ ⎜ ⎜ ⎜ Qm (t, u, v) := ⎜ ⎜ ⎝ u1 v1 = (u1 , , um ), v = (v1 , , vm ) ∈ Rm ⎞ u1 v1 ⎟ ⎟ ⎟ um vm ⎟ ⎟ um t ⎠ vm t 180 OLEG R MUSIN Let Dm := {(t, u, v) ∈ R2m+1 : Qm (t, u, v) 0} Since det(Qm (t, u, v)) = (1 − |u|2 )(1 − |v|2 ) − (t − u, v )2 , we have (t, u, v) ∈ Dm if and only if (t − u, v )2 ≤ (1 − |u|2 )(1 − |v|2 ), Denote (det(Qm (t, u, v))) (n−m−3)/2 |u| ≤ by ρn,m (t, u, v), i.e ρn,m (t, u, v) := (1 − |u|2 )(1 − |v|2 ) − (t − u, v )2 (n−m−3)/2 Theorem 2.4 Let ≤ m ≤ n − Let q(u, v) be any continuous function on {(u, v) ∈ R2m : |u| ≤ 1, |v| ≤ 1} If k = , then (n,m) Dm Gk (t, u, v) G (n,m) (t, u, v) q(u, v) ρn,m(t, u, v) dt du dv = Proof Let t = s ((1 − |u|2 )(1 − |v|2 ))1/2 + u, v In the variables s, u, v we have Dm = {(s, u, v) : −1 ≤ s ≤ 1, |u| ≤ 1, |v| ≤ 1} The Jacobian of this change of variables is ((1 − |u|2 )(1 − |v|2 ))1/2 Then (n,m) I= Dm Gk (t, u, v) G = I0 I0 = |u|≤1 −1 (n−m) Gk |v|≤1 (n,m) (s) G (t, u, v) q(u, v) ρn,m(t, u, v) dt du dv (n−m) (s) (1 − s2 )(n−m−3)/2 ds, (1 − |u|2 )(1 − |v|2 ) (n−m+k−2)/2 q(u, v) du dv Thus, (2.1) yields I = Let e1 , , en be an orthonormal basis of Rn , and let (x1 , , xn ) be the coordinates expression of a point x ∈ Rn in this basis Then x(m) as well as (n,m) (n,m) Gk ( x, y , x, y) = Gk ( x, y , x(m) , y (m) ) for x, y ∈ Rn are well defined Lemma 2.1 For any continuous F on Dm and ≤ m ≤ n − we have F x, y , x(m) , y (m) dωn (x) dωn (y) (Sn−1 )2 = ωn−m ωn−m−1 m−1 F (t, u, v) rm (u) rm (v) ρn,m (t, u, v) dt du dv, Dm (i) 1/2 for m > 1, r0 = r1 = 1, and ωn is the surface where rm (u) = i=1 (1 − |u | ) area of Sn−1 for the standard measure dωn Proof Let x, y ∈ Sn−1 Let t = x, y , u = x(m) , v = y (m) Let x−u y−v a= , b= , s = a, b , where h(u) = − |u|2 h(u) h(v) Then F x, y , x(m) , y (m) = F (h(u) h(v) a, b + u, v , u, v) = F (t, u, v) MULTIVARIATE POSITIVE DEFINITE FUNCTIONS ON SPHERES 181 Therefore, we can change the variables (x, y) → (a, b, u, v) → (s, u, v) → (t, u, v) We obviously have |u| ≤ 1, |v| ≤ 1, −1 ≤ s ≤ Clearly, |a| = |b| = Then (a, b) ∈ (Sn−m−1 )2 Since |t − u, v | ≤ h(u) h(v) we have (t, u, v) ∈ Dm Consider (x, y) → (a, b, u, v) The Jacobian is defined by dωn (x) = kn,m (u) du dωn−m (a), m−1 kn,m (u) = (1 − |u|2 )(n−m−2)/2 (1 − |u(i) |2 )1/2 i=0 ) : a, b = s} and the Jacobian of Combining the volume of {(a, b) ∈ (S (a, b, u, v) → (s, u, v) we get: ωn−m ωn−m−1 ρn,m (s, 0, 0) The Jacobian of the changing (s, u, v) → (t, u, v) is (h(u) h(v))−1 That completes the proof n−m−1 Combining Lemma 2.1 and Theorem 2.1 we get: Corollary 2.1 Let ≤ m ≤ n − Let f (u, v) be any continuous function on {(u, v) ∈ R2m : |u| ≤ 1, |v| ≤ 1} If k = , then (n,m) (Sn−1 )2 Gk ( x, y , x, y) G (n,m) ( x, y , x, y) f x(m) , y (m) dωn (x) dωn (y) = 2.4 The addition theorem Theorem 2.5 (The addition theorem) Let ≤ m ≤ n − Then k (n,m−1) (t, u, v) Gk n,m n,m Ck−s (u) Ck−s (v) Gs(n,m) (t, u, v), = s=0 where Cdn,m (u) = Cdn,m (u1 , , um ) is a polynomial in u1 , , um of degree d Proof Suppose t, u = (u1 , , um ), v = (v1 , , vm ) are such that |u| < 1, (t − u, v )2 < (1 − |u|2 )(1 − |v|2 ) |v| < 1, Then θ1 , θ2 , ϕ ∈ (0, π) are uniquely defined by the following equations: um vm cos θ1 = , cos θ2 = , 2 2 − u1 − − um−1 − v1 − − vm−1 t − u, v (1 − |u|2 )(1 − |v|2 ) cos ϕ = The addition theorem for Gegenbauer polynomials yields: k (n,m−1) Gk n,m n,m Ck−s (u) Ck−s (v) Gs(n,m) (t, u, v), (t, u, v) = (2.2) s=0 √ um , w = − u21 − − u2m−1 w n,m It’s easy to see that Ck−s (u1 , , um ) is a polynomial of degree k − s Thus, (2.2) holds for all t, u, v n,m Ck−s (u) := (n+2s) cnks wk−s Gk−s n,m n,m Let bs (u, v) = Ck−s (u) Ck−s (v) Clearly, bs Then Theorem 2.2 yields 182 OLEG R MUSIN Corollary 2.2 Let ≤ m ≤ ≤ n − Let u, v ∈ Rn Then k (n,m) Gk fs u( ) , v( (t, u, v) = ) ) G(n, (t, u, v) s s=0 with fs for all ≤ s ≤ k An extension of the Schoenberg theorem 3.1 Schoenberg’s theorem Let p1 , , pr be points in Sn−1 , and let a1 , , ar be any real numbers Then ≤ |a1 p1 + + ar pr | = p i , p j aj , i,j or equivalently the Gram matrix pi , pj is positive semidefinite (n) Schoenberg [21] extended this property to Gegenbauer polynomials Gk He (n) proved that for any finite X = {p1 , , pr } ⊂ Sn−1 the matrix Gk ( pi , pj ) is positive semidefinite Schoenberg proved also that the converse holds: if f (t) is a real polynomial 0, then f (t) is a linear and for any finite X ⊂ Sn−1 the matrix f ( pi , pj ) (n) combination of Gk (t) with nonnegative coefficients 3.2 An extension of the direct Schoenberg theorem Theorem 3.1 Let e1 , , en be an orthonormal basis of Rn , and let p1 , , pr be points in Sn−1 Then for any k ≥ and ≤ m ≤ n − the matrix (n,m) Gk ( pi , pj , pi , pj ) is positive semidefinite Proof Actually, this theorem is a simple consequence of the Schoenberg theorem Indeed, let vi = pi , e1 e1 + + pi , em em , and let xi = pi − vi Then xi is a vector in Rn−m with the basis em+1 , , en Note that |vi |2 + |xi |2 = Let yi = xi /|xi | for |xi | > In the case |xi | = put yi = en Then yi ∈ Sn−m−1 0, B = bij 0, then C = Recall the Schur theorem: If A = aij aij bij Let hi = |xi |k = (1 − |vi |2 )k/2 , h = (h1 , , hr ) , A = hT h Clearly, A (n−m) The Schoenberg theorem yields: B = Gk is easy to see that (n,m) Gk (tij , vi , vj ) (n,m) Gk (tij , vi , vj ) ( yi , yj ) It = aij bij , where tij := pi , pj Thus, C = (2) Using the addition theorem it is not hard to give a direct proof Note that Gk (2) is the Chebyshev polynomial of the first kind, i.e Gk (cos θ) = cos(kθ) It is easy From this follows to see that for any ϕ1 , , ϕr the matrix cos(ϕi − ϕj ) that Schoenberg’s theorem holds for n = Therefore, we have proved the theorem for m = n − Put = n − in Corollary 2.2 That yields the theorem for all ≤ m ≤ n − Remark 3.1 We see that the second proof (as well as Schoenberg’s original proof in [21]) is based on the addition theorem for Gegenbauer polynomials There exists another proof of Schoenberg’s theorem which is using the addition theorem MULTIVARIATE POSITIVE DEFINITE FUNCTIONS ON SPHERES 183 for spherical harmonics (see, for instance, [20]) It is possible (see [2] for the case m = 1) to derive Theorem 3.1 from this theorem However, this proof looks more complicated than our proof Remark 3.2 Bachoc and Vallentin [2] derived new upper bounds for spherical codes based on positive semidefinite constraints that are given in [2, Corollary 3.4] In fact, this corollary easily follows from Theorem 3.1 with m = Indeed, let us fix some , ≤ ≤ r, and take an orthonormal basis e1 , , en of Rn with e1 = p Then for m = we have vi = ti Let (n,1) (A )ij = Gk (tij , ti , tj ), ≤ i ≤ r, ≤ j ≤ r, (n+2k) (tj ), ≤ i ≤ d, ≤ j ≤ r Y = W A W T , (W )ij = λi Gi Theorem 3.1 yields: A Then Y and Y1 + + Yr These constraints are equivalent to the constraints in [2, Corollary 3.4] Corollary 3.1 Let ≤ m ≤ n − 2, d ≥ Let e1 , , en be an orthonormal basis of Rn Let a polynomial F (t, u, v) can be represented in the form d (n,m) F (t, u, v) = fk (u, v) Gk (t, u, v), k=0 where fk for all ≤ k ≤ d Then for any points p1 , , pr in Sn−1 the (m) (m) matrix F pi , pj , pi , pj is positive semidefinite In other words, F ∈ PD(Sn−1 , {e1 , , em }) From Proof Let f (u, v) Then f (u, v) = H, Zdm (u, v) with H an eigenvalue factorization of H it follows that there exist polynomials hi (u) such (m) (m) that f (u, v) = i hi (u) hi (v) Therefore, f pi , pj (m) We have Ak = fk pi (m) , pj Theorem 3.1 yields (n,m) (m) pi , pj , pi Bk = G k (m) , pj Let (Ck )ij = (Ak )ij (Bk )ij The Schur theorem implies: Ck (m) F pi , pj , pi (m) , pj Thus, d = Ck k=0 3.3 An extension of the converse Schoenberg theorem Theorem 3.2 Let ≤ m ≤ n − Let e1 , , en ∈ Rn be an orthonormal basis Let F (t, u, v) be a polynomial in t and a symmetric polynomial in u, v ∈ Rm Suppose F ∈ PD(Sn−1 , {e1 , , em }) Then d (n,m) F (t, u, v) = fk (u, v) Gk (t, u, v) (3.1) k=0 with fk for all ≤ k ≤ d = degt (F ) (n,m) (t, u, v) has degree k in the variable t, so that F Proof The polynomial Gk has a unique expression in the form (3.1), where fk (u, v) is a symmetric polynomial in u, v of degree dk in u Then fk (u, v) = Fk , Zdmk (u, v) , where Fk is a symmetric matrix Now we prove that Fk 184 OLEG R MUSIN Let f (t, u, v), g(t, u, v) be continuous functions on Dm Denote f ( x, y , x(m) , y (m) ) g( x, y , x(m) , y (m) ) dωn (x) dωn (y) f, g := (Sn−1 )2 For any continuous function h(u, v) Corollary 2.1 yields: (n,m) F, h Gk (n,m) = fk Gk (n,m) , h Gk (n,m) = fk h, Gk (3.2) Let f (u, v), g(u, v) be continuous functions on {|u| ≤ 1, |v| ≤ 1} Denote (n,m) [f, g] := f g, Gk Clearly, [·, ·] is an inner product Let α1 , α2 , be an orthonormal basis for [·, ·] in the space of real polynomials R[u, v] We observe that Fk defines a quadratic form on R[u, v] Let us denote by F˜k the matrix expression of this quadratic form in the basis {αi } Then there exists a matrix A such that Fk = AT F˜k A Therefore, F˜k yields Fk ˜ Let h(u, v) = H, Zdmk (u, v) It’s easy to see that [fk , h] = F˜k , H Note that the sum of all entries of a positive semidefinite matrix is nonnegative This implies: if f ∈ PD(Sn−1 , Em ), Em := {e1 , , em }, then f, ≥ Moreover, the Schur theorem yields: if f and g ∈ PD(Sn−1 , Em ), then f g ∈ PD(Sn−1 , Em ) Therefore, f, g ≥ Thus, for any f, g ∈ PD(Sn−1 , Em ) we have [f, g] ≥ Let H Then h(u, v) ∈ PD(Sn−1 , Em ) We have F ∈ PD(Sn−1 , Em ) Then ˜ ≥ for any H ˜ Thus, F˜k and Fk (3.2) yields [fk , h] = F˜k , H Remark 3.3 In [3, Proposition 4.12] a statement that is equivalent to Theorem 3.2 was proven with m = However, in [3, Proposition 4.12] the formula (3.1) is written in terms of an orthogonal basis for ·, · 3.4 The class PD(Sn−1 , Q) If fr ∈ PD(M, Q), fr → f as r → ∞, and f (t, u, v) is continuous, then also f ∈ PD(M, Q) Therefore, Corollary 3.1 and Theorem 3.2 imply Theorem 3.3 Let ≤ m ≤ n − Let e1 , , en be an orthonormal basis of Rn Then F ∈ PD(Sn−1 , {e1 , , em }) if and only if ∞ F (t, u, v) = (n,m) fk (u, v) Gk (t, u, v), k=0 where for any k ≥ a function fk (u, v) is positive semidefinite Now we consider PD(Sn−1 , Q) for any Q = {q1 , , qm } ⊂ Sn−1 with ≤ m ≤ n − Theorem 3.3 yields Corollary 3.2 Let ≤ m ≤ n − Let Q = {q1 , , qm } ⊂ Sn−1 with rank(Q) = m Let e1 , , em be an orthonormal basis of the linear space with the basis q1 , , qm , and let LQ denotes the linear transformation of coordinates Then F ∈ PD(Sn−1 , Q) if and only if ∞ k=0 where fk (u, v) (n,m) fk (u, v) Gk F (t, u, v) = for all k ≥ (t, LQ (u), LQ (v)), MULTIVARIATE POSITIVE DEFINITE FUNCTIONS ON SPHERES 185 Remark 3.4 It is not hard to describe PD(Sn−1 , Q) with rank(Q) ≥ n − First, consider the case Q = {e1 , , en−1 }, where {ei } is an orthonormal basis Let p1 , , pr ∈ Sn−1 , ui = pi , e1 e1 + + pi , en−1 en−1 , and tij = pi , pj Obviously, tij − ui , uj = wi wj , wi = pi , en Then tij − ui , uj = wT w 0, w = (w1 , , wr ) (3.3) Since wi2 = − |ui |2 , we have for all i = 1, , r, j = 1, , r: (tij − ui , uj )2 = (1 − |ui |2 )(1 − |uj |2 ) (3.4) Let H2 (t, u, v) := (t − u, v ) − (1 − |u| )(1 − |v| ) Since H2 (tij , ui , uj ) = 0, we have F ∈ PD(Sn−1 , {e1 , , en−1 }) if and only if 2 F (t, u, v) = f0 (u, v) + f1 (u, v) (t − u, v ) + R(t, u, v) H2 (t, u, v), where f0 0, f1 0, and R(t, u, v) is any continuous function on Dm Let Q = {q1 , , qn−1 } ⊂ Sn−1 with rank(Q) = n − Let L be a linear transformation of the basis q1 , , qn−1 to an orthonormal basis e1 , , en−1 of Rn−1 Then F ∈ PD(Sn−1 , Q) if and only if F (t, u, v) = f0 (u, v) + f1 (u, v) (t − L(u), L(v) ) + R(t, u, v) H2 (t, L(u), L(v)), with f0 0, f1 0, and any R(t, L(u), L(v)) ∈ C(Dm ) In the case rank(Q = {q1 , , qm }) ≥ n−1 consider all Qi ⊂ Q with rank(Qi ) = n − Let a linear transformation Li : Rm → Rn−1 is defined by Li (q) = q if q ∈ Qi , Li (q) = if q ∈ Q and rank(Qi ∪ {q}) = n Denote by N the number of distinct Qi It is not hard to prove that F ∈ PD(Sn−1 , Q) iff N F (t, u, v) = fi (u, v) Fi (t, Li (u), Li (v)), i=1 where for all i: fi (u, v) 0, Fi (t, Li (u), Li (v)) ∈ PD(Sn−1 , Qi ) Positive definite functions in Rn Direct extensions of the Bochner–Schoenberg theorem and finding bounds on sphere packings in Rn are not so straightforward because this space is not compact Different indirect ways of deriving bounds on sphere packings in Rn were suggested in the literature [8, 14, 16] Here we note that the multivariate Gegenbauer polynomials defined above enable one to define p.d functions in Rn as follows: (n,m) Hk (n,m) (t, x, y, u, v) := (xy)k/2 Gk (t , u , v ), where ≤ m ≤ n − 2, t, x, y ∈ R, u, v ∈ R for m > and u = v = for m = 0, and √ √ √ t = t/ xy, u = u/ x, v = v/ y m (n,m) The positive semidefiniteness of the polynomials Gk following result (Theorem 3.1) implies the Theorem 4.1 Let e1 , , en ∈ Rn be an orthonormal basis, and let p1 , , pN be points in Rn Then for any k ≥ and ≤ m ≤ n − the matrix gij , where (n,m) gij = Hk is positive semidefinite ( pi , pj , |pi |2 , |pj |2 , pi , pj ), 186 OLEG R MUSIN Remark 4.1 It is not hard to find Euclidean analogs of Theorems 3.2, 3.3 and Corollary 3.2 Theorem 4.1 gives a family of positive-semidefinite constraints for distance distributions of points in Euclidean spaces For instance, consider the simple case (n) of m = Now for any matrix A = aij of size N × N we have a matrix Hk (A) which is defined by √ (n) (n) = (aii ajj )k/2 Gk aij / aii ajj Hk (A) ij Corollary 4.1 If Ais a symmetric positive semidefinite matrix with rank(A)≤ n, then for any positive integer k we have (n) Hk (A) Proof It is a well-known fact: A symmetric matrix A of size N × N is the Gram matrix of N vectors in Rn if and only if A and rank(A) ≤ n Therefore, there are vectors p1 , , pN in Rn such that A = pi , pj Then (n) Hk (A) (n,0) ij = Hk (n) Thus Theorem 4.1 yields Hk (A) ( pi , pj , |pi |2 , |pj |2 , 0, 0) (n) For small k it is not hard to give explicit expressions for Hk (A) Clearly, (n) = A Since G2 (t) = (nt2 − 1)/(n − 1), we have (n) H1 (A) (n) H2 (A) ij = Then (n) H2 (A) = na2ij − aii ajj n−1 nA2 − aT a , n−1 where (A2 )ij := a2ij , So if A a := (a11 , a22 , , aN N ) and rank(A) ≤ n, then nA2 aT a Upper bounds for spherical codes In this section we set up upper bounds for spherical codes which are based on multivariate p.d functions These bounds extend the famous Delsarte’s bound Note that for the case m = this bound is the Bachoc - Vallentin bound [4] Definition 5.1 Consider a vector J = (j1 , , jd ) Split the set of numbers {j1 , , jd } into maximal subsets I1 , , Ik with equal elements That means, if Ir = {jr1 , , jrs }, then jr1 = = jrs = ar and all other j = ar Without loss of generality it can be assumed that i1 = |I1 | ≥ ≥ ik = |Ik | > (Note that we have i1 + + ik = d.) Denote by ψ(J) the vector ω = (i1 , , ik ) Let Wd := {ω = (i1 , , ik ) : i1 + + ik = d, i1 ≥ ≥ ik > 0, i1 , , ik ∈ Z} Let ω ∈ Wd Denote q˜ω (N ) := #{J = (j1 , , jd ) ∈ {1, , N }d : ψ(J) = ω}, MULTIVARIATE POSITIVE DEFINITE FUNCTIONS ON SPHERES 187 q˜ω (N ) N It is not hard to see that qω (N ) is a polynomial of degree d − for ω ∈ Wd and qω (N ) := qω (N ) = N d−1 ω∈Wd Definition 5.2 For any vector x = {xij } with ≤ i < j ≤ d denote by A(x) a symmetric d × d matrix aij with all aii = and aji = aij = xij , i < j Let < θ < π and X(θ) := {x = {xij } : xij ∈ [−1, cos θ] or xij = 1, ≤ i < j ≤ d} Now for any x = {xij } ∈ X(θ) we define a vector J(x) = (j1 , , jd ) such that jk = k if there are no i < k with xik = 1, otherwise jk = i, where i is the minimum index with xik = Let ω ∈ Wd Denote Dω (θ) := {x ∈ X(θ) : ψ(J(x)) = ω and A(x) 0} Let f (x) be a real function in x, and let Bω (θ, f ) := sup f (x) x∈Dω (θ) Note that the assumption A(x) implies existence of unit vectors p1 , , pd such that A(x) is the Gram matrix of these vectors, i.e xij = pi , pj Moreover, if xij = 1, then pi = pj In particular, D(d) (θ) = {(1, , 1)} and therefore B(d) (θ, f ) = f (1, , 1) Definition 5.3 Let x = {xij }, where ≤ i < j ≤ m+2 ≤ n, and let A(x) Then there exist P = {p1 , , pm+2 } ⊂ Sn−1 such that xij = pi , pj Let F (x) be a continuous function in x with F (˜ xk ) = F (x) for all x ˜k that can be obtained by interchanging two points pk and p in P We say that F (x) ∈ PDnm if for all x with A(x) we have F˜ (x12 , u1 , u2 ) ∈ PD(Sn−1 , Q(x)), where ui = (xi3 , , xi,m+2 ), Q(x) = {p3 , , pm+2 }, and F˜ (x12 , u1 , u2 ) = F (x) For the classical case m = Schoenberg’s theorem says that f ∈ PDn0 if and (n) only if f (t) = k fk Gk (t) with all fk ≥ Theorem 3.3 (see also [2–4]) yields n F (x) ∈ PD1 if and only if (n,1) F (x12 , x13 , x23 ) = fk (x12 , x13 , x23 ) Gk (x12 , x13 , x23 ), k where fk for all k, and F (x12 , x13 , x23 ) = F (x13 , x12 , x23 ) = F (x23 , x13 , x12 ) Using Corollary 3.2 it is possible to describe the class of functions PDnm , m ≥ Let C be an N -element subset of the unit sphere Sn−1 ⊂ Rn It is called an (n, N, θ) spherical code if every pair of distinct points (c, c ) of C have inner product c, c at most cos θ Theorem 5.4 Let f0 > 0, ≤ m ≤ n − 2, and F (x) = f (x) − f0 ∈ PDnm Then an (n, N, θ) spherical code satisfies f0 N m+1 ≤ Bω (θ, f ) qω (N ) ω∈Wm+2 188 OLEG R MUSIN Proof Let C be an (n, N, θ) spherical code Define f ({ ci , cj }), S= c=(c1 , ,cd d = m + )∈C d Then f ({ ci , cj }) ≤ S= ω∈Wd c:ψ(c)=ω Bω (θ, f ) q˜ω (N ) ω∈Wd On the other hand, since F ∈ PDnm we have F ({ ci , cj }) ≥ c∈C d Thus (f0 + F ({ ci , cj })) ≥ f0 N d S= c∈C d It is easy to see for m = that q(2) (N ) = 1, q(1,1) (N ) = N −1, and B(2) (θ, f ) = f (1) Therefore, from Theorem 5.1 we have f0 N ≤ f (1) + B(1,1) (θ, f )(N − 1) Suppose B(1,1) (θ, f ) ≤ 0, i.e f (t) ≤ for all t ∈ [−1, cos θ] Thus for (n, N, θ) spherical code we obtain f (1) N≤ f0 This upper bound is called Delsarte’s bound The Bachoc-Vallentin bound [4, Theorem 4.1] is the bound in Theorem 5.1 for m = and B(1,1,1) (θ, f ) ≤ Indeed, let B(2,1) (θ, f ) ≤ B Since q(3) (N ) = 1, q(2,1) (N ) = 3(N − 1), and B(3) (θ, f ) = f (1, 1, 1), we have f0 N ≤ f (1, 1, 1) + 3(N − 1)B Let us consider Theorem 5.1 also for the case m = with B(1,1,1,1) (θ, f ) ≤ Let B(3,1) (θ, f ) ≤ B1 , B(2,2) (θ, f ) ≤ B2 , and B(2,1,1) (θ, f ) ≤ B3 Then f0 N ≤ f (1, 1, 1, 1, 1, 1) + 4(N − 1)B1 + 3(N − 1)B2 + 6(N − 1)(N − 2)B3 Let f (x) be a polynomial of degree d Then the assumptions in Theorem 5.1 can be written as positive semidefinite constraints for the coefficients of F (see for details [2–4, 13, 22]) Actually, the bound given by Theorem 5.1 can be obtained as a solution of an SDP (semidefinite programming) optimization problem In [2, 3] using numerical solutions of the SDP problem for the case m = has obtained new upper bounds for the kissing numbers and for the one-sided kissing numbers in several dimensions n ≤ 10 However, the dimension of the corresponding SDP problem is growth so fast whenever d and m are increasing that this problem can be treated numerically only for relatively small d and small m It is an interesting problem to find (explicitly) suitable polynomials F for Theorem 5.1 and using it to obtain new bounds for spherical codes MULTIVARIATE POSITIVE DEFINITE FUNCTIONS ON SPHERES 189 References [1] C Bachoc, Semidefinite programming, harmonic analysis and coding theory, preprint, arXiv:0909.4767, 2009 [2] Christine Bachoc and Frank Vallentin, New upper bounds for kissing numbers from semidefinite programming, J Amer Math Soc 21 (2008), no 3, 909–924, DOI 10.1090/S0894-034707-00589-9 MR2393433 (2009c:52029) [3] Christine Bachoc and Frank Vallentin, Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps, European J Combin 30 (2009), no 3, 625–637, DOI 10.1016/j.ejc.2008.07.017 MR2494437 (2010d:90065) [4] Christine Bachoc and Frank Vallentin, 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programming, Tr Mat Inst Steklova 263 (2008), no Geometriya, Topologiya i Matematicheskaya Fizika I, 143–158, DOI 10.1134/S0081543808040111; English transl., Proc Steklov Inst Math 263 (2008), no 1, 134–149 MR2599377 (2011c:94085) [20] Florian Pfender and Gă unter M Ziegler, Kissing numbers, sphere packings, and some unexpected proofs, Notices Amer Math Soc 51 (2004), no 8, 873–883 MR2145821 (2006a:52015) [21] I J Schoenberg, Positive definite functions on spheres, Duke Math J (1942), 96–108 MR0005922 (3,232c) 190 OLEG R MUSIN [22] Alexander Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans Inform Theory 51 (2005), no 8, 2859–2866, DOI 10.1109/TIT.2005.851748 MR2236252 (2007a:94147) Department of Mathematics, University of Texas at Brownsville, Brownsville, Texas 78520 — and — Inst for Problems of Inform Trans., RAS, Moscow, Russia E-mail address: oleg.musin@utb.edu Selected Published Titles in This Series 625 Alexander Barg and Oleg R Musin, Editors, Discrete Geometry and Algebraic Combinatorics, 2014 616 G L Litvinov and S N Sergeev, Editors, Tropical and Idempotent Mathematics and Applications, 2014 615 Plamen Stefanov, Andr´ as Vasy, and Maciej Zworski, Editors, Inverse Problems and Applications, 2014 614 James W Cogdell, Freydoon Shahidi, and David Soudry, Editors, Automorphic Forms and Related Geometry, 2014 613 Stephan Stolz, Editor, Topology and Field Theories, 2014 612 Patricio Cifuentes, Jos´ e Garc´ıa-Cuerva, Gustavo Garrig´ os, Eugenio Hern´ andez, Jos´ e Mar´ıa Martell, Javier Parcet, Keith M Rogers, Alberto Ruiz, Fernando Soria, and Ana Vargas, Editors, Harmonic Analysis and Partial Differential Equations, 2014 611 Robert Fitzgerald Morse, Daniela Nikolova-Popova, and Sarah Witherspoon, Editors, Group Theory, Combinatorics, and Computing, 2014 610 Pavel Etingof, Mikhail Khovanov, and Alistair Savage, Editors, Perspectives in Representation Theory, 2014 609 Dinh Van Huynh, S K Jain, Sergio R L´ opez-Permouth, S Tariq Rizvi, and Cosmin S Roman, Editors, Ring Theory and Its Applications, 2014 608 Robert S Doran, Greg Friedman, and Scott Nollet, Editors, Hodge Theory, Complex Geometry, and Representation Theory, 2014 607 Kiyoshi Igusa, Alex Martsinkovsky, and Gordana Todorov, Editors, Expository Lectures on Representation Theory, 2014 606 Chantal David, Matilde Lal´ın, and Michelle Manes, Editors, Women in Numbers 2, 2013 605 Omid Amini, Matthew Baker, and Xander Faber, Editors, Tropical and Non-Archimedean Geometry, 2013 604 Jos´ e Luis Monta˜ na and Luis M Pardo, Editors, Recent Advances in Real Complexity and Computation, 2013 ´ 603 Azita Mayeli, Alex Iosevich, Palle E T Jorgensen, and Gestur Olafsson, Editors, Commutative and Noncommutative Harmonic Analysis and Applications, 2013 602 Vyjayanthi Chari, Jacob Greenstein, Kailash C Misra, K N Raghavan, and Sankaran Viswanath, Editors, Recent Developments in Algebraic and Combinatorial Aspects of Representation Theory, 2013 601 David Carf`ı, Michel L Lapidus, Erin P J Pearse, and Machiel van Frankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II, 2013 600 David Carf`ı, Michel L Lapidus, Erin P J Pearse, and Machiel van Frankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I, 2013 599 Mohammad Ghomi, Junfang Li, John McCuan, Vladimir Oliker, Fernando Schwartz, and Gilbert Weinstein, Editors, Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, 2013 598 Eric Todd Quinto, Fulton Gonzalez, and Jens Gerlach Christensen, Editors, Geometric Analysis and Integral Geometry, 2013 597 Craig D Hodgson, William H Jaco, Martin G Scharlemann, and Stephan Tillmann, Editors, Geometry and Topology Down Under, 2013 596 Khodr Shamseddine, Editor, Advances in Ultrametric Analysis, 2013 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/conmseries/ CONM 625 ISBN 978-1-4704-0905-0 AMS 781470 409050 CONM/625 Discrete Geometry and Algebraic Combinatorics • Barg et al., Editors This volume contains the proceedings of the AMS Special Session on Discrete Geometry and Algebraic Combinatorics held on January 11, 2013, in San Diego, California The collection of articles in this volume is devoted to packings of metric spaces and related questions, and contains new results as well as surveys of some areas of discrete geometry This volume consists of papers on combinatorics of transportation polytopes, including results on the diameter of graphs of such polytopes; the generalized Steiner problem and related topics of the minimal fillings theory; a survey of distance graphs and graphs of diameters, and a group of papers on applications of algebraic combinatorics to packings of metric spaces including sphere packings and topics in coding theory In particular, this volume presents a new approach to duality in sphere packing based on the Poisson summation formula, applications of semidefinite programming to spherical codes and equiangular lines, new results in list decoding of a family of algebraic codes, and constructions of bent and semi-bent functions ... www.TechnicalBooksPDF.com www.TechnicalBooksPDF.com 625 Discrete Geometry and Algebraic Combinatorics AMS Special Session Discrete Geometry and Algebraic Combinatorics January 11, 2013 San Diego, CA Alexander... Data Discrete geometry and algebraic combinatorics / Alexander Barg, Oleg R Musin, editors pages cm – (Contemporary mathematics ; volume 625) “AMS Special Session on Discrete Geometry and Algebraic. . .Discrete Geometry and Algebraic Combinatorics AMS Special Session Discrete Geometry and Algebraic Combinatorics January 11, 2013 San Diego, CA Alexander Barg Oleg R Musin