Realizability of Polytopes as a Low Rank Matrix Completion Problem

TL;DR

This paper establishes conditions for polytope realizability through matrix parametrizations, linking combinatorial structures to geometric and algebraic properties in polytope theory.

Contribution

It provides necessary and sufficient conditions for polytope realizability and introduces matrix parametrizations for the moduli spaces of combinatorial polytopes.

Findings

Conditions for facet-vertex relations in polytopes

Matrix parametrizations of moduli spaces

Link between combinatorial and geometric properties

Abstract

This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope.