Perfect, strongly eutactic lattices are periodic extreme

TL;DR

This paper proves that perfect, strongly eutactic lattices are locally optimal for sphere packing density among periodic packings, extending known optimality results to a broader class of packings.

Contribution

It introduces a parameter space for periodic point sets and establishes conditions under which perfect, strongly eutactic lattices are locally optimal for sphere packing density.

Findings

Perfect, strongly eutactic lattices are locally optimal for periodic sphere packings.

The results apply to the densest known lattice packings in dimensions up to 8 and 24.

Periodic improvements cannot surpass these lattices in density.

Abstract

We introduce a parameter space for periodic point sets, given as unions of $m$ translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension $d\leq 8$ and $d=24$.