Closure of principal L-type domain and its parallelotopes

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.