Inhomogeneous extreme forms

TL;DR

This paper bridges classical lattice reduction theories for sphere packing and covering, classifies local covering maxima up to dimension 6, and reveals new phenomena contrasting packing problems.

Contribution

It introduces a novel perspective on lattice coverings by analyzing uneconomical coverings, classifies local maxima up to dimension 6, and demonstrates the existence of such maxima in higher dimensions.

Findings

Many symmetric lattices give uneconomical coverings

Covering density function is not a Morse function

Local covering maxima classified up to dimension 6

Abstract

G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs. By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension 6 and prove their existence in all dimensions beyond. New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast to the packing problem.