2002/03/13 by Michel Deza, Deza, Michel, Viacheslav Grishukhin +1 · 1 citation
Mathematics · #05B45 #52B12 #Analytic and geometric function theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Primary 52B22 #Secondary 05B35 #math.CO #math.MG #msc:05B35 #msc:05B45 #msc:52B12 #msc:52B22
paper · pdf · doi:10.48550/arxiv.math/0203124
10 pages, no figures
arxiv created 2002/03/13 · openalex publication_date 2002/03/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Voronoi conjectured that any parallelotope is affinely equivalent to a Voronoi polytope. A parallelotope is defined by a set of m facet vectors pi and defines a set of m lattice vectors ti, 1≤ i≤ m. We show that Voronoi's conjecture is true for an n-dimensional parallelotope P if and only if there exist scalars γi and a positive definite n× n matrix Q such that γi pi=Qti for all i. In this case the quadratic form f(x)=xTQx is the metric form of P. As an example, we consider in detail the case of a zonotopal parallelotope. We show that Q=(ZβZTβ)-1 for a zonotopal parallelotope P(Z) which is the Minkowski sum of column vectors zj of the n× r matrix Z. Columns of the matrix Zβ are the vectors √(2βj)zj, where the scalars βj, 1≤ j≤ r, are such that the system of vectors \βjzj:1≤ j≤ r\ is unimodular. P(Z) defines a dicing lattice which is the set of intersection points of the dicing family of hyperplanes H(j,k)=\x:xT(βjQzj)=k\, where k takes all integer values and 1≤ j≤ r.