Closure of principal L-type domain and its parallelotopes

2013/04/09 by Sikiric, Mathieu Dutour, Grishukhin, Viacheslav
#05B45 #52B11 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1304.2498

Abstract

Voronoi defined two polyhedral partitions of the cone of se\mi\de\fi\nite forms into L-type domains and into perfect domains. Up to equivalence, there is only one domain that is simultaneously perfect and L-type. Voronoi called this domain \em principal. We show that closure of the principal domain may be identified with a cone of cut submodular set functions. Parallelotopes of the closed principal domain are zonotopes that are base polyhedra related to graphic unimodular sets of vectors.

Related

• The decomposition of the hypermetric cone into L-domains
• Voronoi-Dickson Hypothesis on Perfect Forms and L-types
• Quadratic forms of rank 1, closed zones and laminae
• Properties of parallelotopes equivalent to Voronoi's conjecture
• On the sum of the Voronoi polytope of a lattice with a zonotope