Lower bounds for sphere packing in arbitrary norms

TL;DR

This paper establishes improved lower bounds for sphere packing densities in any finite-dimensional normed space, extending previous results and employing graph theory and convex geometry techniques.

Contribution

It provides a new lower bound on sphere packing density in arbitrary norms, improving upon Schmidt's 1958 result by a logarithmic factor and generalizing recent $ ext{l}_2$ norm results.

Findings

Lower bounds for packing density involve a factor of rac{(1-o(1))d ext{log} d}{2^{d+1}}.

The approach combines graph-theoretic methods with convex geometry volume bounds.

Results apply to any $d$-dimensional real normed space.

Abstract

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of Campos, Jenssen, Michelen, and Sahasrabudhe in the $\ell_2$ norm. Our main tools are the graph-theoretic result used in the $\ell_2$ construction and volume bounds from convex geometry due to Petty and Schmuckenschl\"ager.