Strict Periodic Extreme Lattices

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.

arXiv comment6 pages; to appear in AMS Contemporary Mathematics issue on "Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms", edited by Wai Kiu Chan, Lenny Fukshansky, Rainer Schulze-Pillot, and Jeff Vaaler