Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices
Robert Erdahl, Andrei Ordine, Konstantin Rybnikov
Abstract
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.
Create a lesson
Related papers
An Infinite Family of Apéry-like Recurrences for Euler's Constant
Rotem Kalisch, Tali Monderer, Elyasheev Leibtag et al.
On the saturability of p-adic Lie groups
Casey Perdue
On Divisor Function Polynomials and Dirichlet Series
Martin N. Huxley, Matthew C. Lettington, Karl Michael Schmidt
On a conjecture on Romanoff type representations
Xiao-Hui Yan
Regularization of a Divergent Arithmetic Product via the Asymptotic Monoid: The Case of (-1)ω(n)
A. Alvarez Cruz, E. A. Alvarez Gutierrez
Appell Sequences in Analytic Number Theory: Shifted Expansions and Dirichlet L-Functions
Abdelmejid Bayad