Facet volumes of polytopes

TL;DR

This paper characterizes the space of facet volume vectors of d-polytopes with n facets as a full-dimensional cone, using topological and differential geometric methods, with implications for understanding polytope realization spaces.

Contribution

It introduces a novel topological and geometric approach to describe the configuration space of facet volumes of polytopes, extending previous work and opening new research directions.

Findings

Configuration space of facet volumes is a full-dimensional cone.

For tetrahedra, the cone is over a regular octahedron.

Method combines topological and differential geometric tools.

Abstract

In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers $d\geq 2$ and $n\geq d+1$ the configuration space of all facet volume vectors of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets is a full dimensional cone in $\mathbb R^{n}$. In particular, for tetrahedra ($d=3$ and $n=4$) this is a cone over a regular octahedron. Our proof is based on a novel configuration space / test map scheme which uses topological methods for finding solutions of a problem, and tools of differential geometry to identify solutions with the desired properties. Furthermore, our results open a possibility for the study of realization spaces of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets by the methods of algebraic topology.