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Discrete Geometry and Algebraic Combinatorics 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 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 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) 1. Discrete geometry–Congresses. 2. Combinatorial analysis–Congresses. I. Barg, Alexan- der, 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 edu- cational 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 acknowledg- ment 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 Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. 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/ 10987654321 191817161514 Contents Preface vii Plank theorems via successive inradii K. Bezdek 1 Minimal fillings of finite metric spaces: The state of the art A. Ivanov and A. Tuzhilin 9 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 H. Cohn, A. Kumar, C. Reiher, and A. Schurmann¨ 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 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 Combina- torics” (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 viii PREFACE metric trees. They also exhibit connections to Knaster-Kuratowski-Mazurkiewicz- type theorems and discuss interesting applications to voting theory. In the second area (algebraic combinatorics), A. Barg and W.-H. Yu use semi- definite programming to obtain new bounds on the maximum cardinality of equian- gular 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. Sch¨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 ob- tained 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 program- ming bounds on spherical codes. Previously these bounds were derived using posi- tive 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 polyno- mials satisfy the addition formula as well as positivity conditions generalizing the conditions given the classical Schoenberg theorem for univariate Gegenbauer poly- nomials. 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 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
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