Sphere packing bounds via rescaling

TL;DR

This paper investigates the connection between local and global sphere packing densities, introducing a framework of packing bound functions and a hierarchy of bounds that converge to the optimal density.

Contribution

It formalizes the concept of packing bound functions, generalizes convergence results, and develops a Lasserre hierarchy for improved bounds on sphere packing density.

Findings

Packing bound functions tend to a Euclidean limit on rectifiable sets.

Linear and semidefinite programming bounds can be formulated as packing bound functions.

A Lasserre hierarchy converges to the optimal sphere packing density.

Abstract

We study the relationship between local and global density for sphere packings, and in particular the convergence of packing densities in large, compact regions to the Euclidean limit. We axiomatize key properties of sphere packing bounds by the concept of a packing bound function, and we study the special case of sandwich functions, which give a framework for inequalities given by the Lov\'asz sandwich theorem. We show that every packing bound function tends to a Euclidean limit on rectifiable sets, generalizing the work of Borodachov, Hardin, and Saff. Linear and semidefinite programming bounds yield packing bound functions, and we develop a Lasserre hierarchy that converges to the optimal sphere packing density.