2006/12/30 by Robert Erdahl, Erdahl, Robert, Andrei Ordine +3
Computer Science · Mathematics · #11-xx #Advanced Combinatorial Mathematics #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT) #Polynomial and algebraic computation #math.MG #math.NT #msc:11-xx #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0701006
24 pages, including 2 figures
arxiv created 2006/12/30 · openalex publication_date 2006/12/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A polytope D whose vertices belong to a lattice of rank d is Delaunay if there is a circumscribing d-dimensional ellipsoid, E, with interior free of lattice points so that the vertices of D lie on E. If in addition, the ellipsoid E is uniquely determined by D, we call D perfect. That is, a perfect Delaunay polytope is a lattice polytope with a circumscribing empty ellipsoid E, where the quadratic surface ∂ E both contains the vertices of D and is determined by them. We have been able to construct 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 play an important role in the theory of Delaunay polytopes, and in Voronoi's theory of lattice types.