Inhomogeneous extreme forms
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.
Keywords
Cite
@article{arxiv.1008.4751,
title = {Inhomogeneous extreme forms},
author = {Mathieu Dutour Sikiric and Achill Schuermann and Frank Vallentin},
journal= {arXiv preprint arXiv:1008.4751},
year = {2014}
}
Comments
22 pages, revision based on suggestions by referee, accepted in Annales de l'Institut Fourier