Perfect Delaunay Polytopes and Perfect Inhomogeneous Forms
Abstract
A lattice Delaunay polytope D is called perfect if it has the property that there is a unique circumscribing ellipsoid with interior free of lattice points, and with the surface containing only those lattice points that are the vertices of D. An inhomogeneous quadratic form is called perfect if it is determined by such a circumscribing ''empty ellipsoid'' uniquely up to a scale factor. Perfect inhomogeneous forms are associated with perfect Delaunay polytopes in much the way that perfect homogeneous forms are associated with perfect point lattices. We have been able to construct some infinite sequences of perfect Delaunay polytopes, one perfect polytope in each successive dimension starting at some initial dimension; we have been able to construct an infinite number of such infinite sequences. Perfect Delaunay polytopes are intimately related to the theory of Delaunay polytopes, and to Voronoi's theory of lattice types.
Keywords
Cite
@article{arxiv.math/0408122,
title = {Perfect Delaunay Polytopes and Perfect Inhomogeneous Forms},
author = {Robert Erdahl and Andrei Ordine and Konstantin Rybnikov},
journal= {arXiv preprint arXiv:math/0408122},
year = {2007}
}
Comments
Release 3: 25 pages, 4 diagrams. A number of errors in notation, terminology, citations, and cross-references have been fixed. Note that in all diagrams kappa should read as k. A reduced 10 page version of this article will apear in the proceedings of the 2003 Voronoi Conference held in Kiev in September of 2003