Discrete Geometry and Algebraic Combinatorics: Ams Special Session Discrete Geometry and Algebraic Combinatorics January 11, 2013 San Diego, Ca

202 Pages·2014·2.2 MB·English

by  Alexander Barg

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
AmericanMathematicalSociety
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
AmericanMathematicalSociety
Providence,RhodeIsland
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
Discretegeometryandalgebraiccombinatorics/AlexanderBarg,OlegR.Musin,editors.
pagescm. –(Contemporarymathematics;volume625)
“AMSSpecialSessiononDiscreteGeometryandAlgebraicCombinatorics,January11,2013.”
Includesbibliographicalreferences.
ISBN978-1-4704-0905-0(alk. paper)
1. Discretegeometry–Congresses. 2. Combinatorialanalysis–Congresses. I. Barg,Alexan-
der,1960-editorofcompilation. II. Musin,O.R.(OlegRustamovich)editorofcompilation.
QA640.7.D5752014
516(cid:2).116–dc23 2014007424
ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online)
DOI:http://dx.doi.org/10.1090/conm/625
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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. Schu¨rmann 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
SanDiego, CA.Thepapersinthevolume belongtooneofthetworelatedsubjects
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
interestfordiscretemathematicsbecausepermutationmatrices,Latinsquares,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.
ThepaperbyA.IvanovandA.Tuzhilinpresentsanoverviewofanewbranchof
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
equivalenttoatheoremforfinitecoversofmetrictreesandafixedpointtheoremon
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-
definiteprogrammingtoobtainnewboundsonthemaximumcardinalityofequian-
gularlinesetsinRn.Theyobtainsomenewexactanswers, resolvinginparta1972
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. Schu¨rmann address the notion of formal duality
introducedearlierintheworkonenergy-minimizingconfigurations. Formalduality
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
fewderivatives. Suchcodeswereknownforabout15yearsinthecaseofunivariate
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-
tivedefinitematrices,whilethispaperdefinesanewclassofmultivariateorthogonal
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
ContemporaryMathematics
Volume625,2014
http://dx.doi.org/10.1090/conm/625/12489
Plank theorems via successive inradii
K´aroly Bezdek
Abstract. Inthe1930’s,Tarskiintroducedhisplankproblematatimewhen
the field discrete geometry was about to born. It is quite remarkable that
Tarski’squestionanditsvariantscontinuetogenerateinterestinthegeometric
aswellas analytic aspectsofcoveringsby planks in the present timeaswell.
Besides giving a short survey on the status of the affine plank conjecture of
Bang(1950)weprovesomenewpartialresultsforthesuccessiveinradiiofthe
convex bodies involved. The underlying geometric structures are successive
hyperplanecutsintroducedseveralyearsagobyConwayandinductivetilings
introducedrecentlybyAkopyanandKarasev.
1. Introduction
Asusual, aconvex bodyoftheEuclideanspaceEd isacompactconvexsetwith
non-emptyinterior. LetC⊂Ed beaconvexbody,andletH ⊂Edbeahyperplane.
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. Aplank PinEd is the(closed)setof pointsbetweentwodistinct parallel
hyperplanes. Thewidthw(P)ofPissimplythedistancebetweenthetwoboundary
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(cid:2)the planks P1,P2,...,Pn in
Ed,d≥2 (i.e., C⊂P ∪P ∪···∪P ⊂Ed), then n w(P )≥w(C).
1 2 n i=1 i
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 MathematicsSubjectClassification. Primary52C17,05B40,11H31,and52C45.
Partially supported by a Natural Sciences and Engineering Research Council of Canada
DiscoveryGrant.
(cid:3)c2014 American Mathematical Society
1