Computing upper bounds for the packing density of congruent copies of a convex body
arXiv:1308.4893 · doi:10.1007/978-3-662-57413-3_7
Abstract
In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in ; this theorem is a generalization of the linear programming bound for sphere packings. We illustrate its use by computing an upper bound for the maximum density of packings of regular pentagons in the plane. Our computational approach is numerical and uses a combination of semidefinite programming, sums of squares, and the harmonic analysis of the Euclidean motion group. We show how, with some extra work, the bounds so obtained can be made rigorous.
(v3) 30 pages, some updates, revision based on suggestions by referee
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