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Upper bounds for packings of spheres of several radii

Part of: Discrete geometry

Published online by Cambridge University Press:  23 September 2014

DAVID DE LAAT ,
FERNANDO MÁRIO DE OLIVEIRA FILHO  and
FRANK VALLENTIN
DAVID DE LAAT
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands; mail@daviddelaat.nl
FERNANDO MÁRIO DE OLIVEIRA FILHO
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, 05508-090 São Paulo, Brazil; fmario@gmail.com
FRANK VALLENTIN
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany; frank.vallentin@uni-koeln.de

Abstract

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We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions 90C22: Semidefinite programming
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e23
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2014

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