Closure of principal L-type domain and its parallelotopes

TL;DR

This paper explores the geometric structure of the principal L-type domain in the cone of semi-definite forms, revealing its closure as a cone of cut submodular functions and describing its parallelotopes as zonotopes linked to graphic unimodular sets.

Contribution

It characterizes the closure of the principal L-type domain as a cone of cut submodular functions and describes its parallelotopes as zonotopes related to graphic unimodular sets.

Findings

Closure of the principal domain is a cone of cut submodular functions.

Parallelotopes of the closed principal domain are zonotopes.

These zonotopes are base polyhedra associated with graphic unimodular sets.

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.