Computing upper bounds for the packing density of congruent copies of a convex body

TL;DR

This paper introduces a generalized linear programming method to compute upper bounds for packing densities of convex bodies, demonstrated through regular pentagon packings, using advanced numerical optimization techniques.

Contribution

It extends the linear programming bound to general convex bodies and provides a computational framework for rigorous upper bound estimation.

Findings

Derived an upper bound for regular pentagon packings in the plane.

Applied semidefinite programming and harmonic analysis for bounds computation.

Showed how to make the bounds rigorous with additional work.

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 $\mathbb{R}^n$; 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.