Strict Periodic Extreme Lattices

TL;DR

This paper characterizes strict periodic extreme lattices as those that are both extreme and non-floating, establishing a precise criterion for local optimality in periodic sphere packings.

Contribution

It provides a complete characterization of strict periodic extreme lattices, linking them to the concepts of extremality and non-floating properties.

Findings

A lattice is periodic extreme if and only if it is extreme.

A lattice is strict periodic extreme if and only if it is extreme and non-floating.

Abstract

A lattice is called periodic extreme if it cannot locally be modified to yield a better periodic sphere packing. It is called strict periodic extreme if its sphere packing density is an isolated local optimum among periodic point sets. In this note we show that a lattice is periodic extreme if and only if it is extreme, that is, locally optimal among lattices. Moreover, we show that a lattice is strict periodic extreme if and only if it is extreme and non-floating.