Facet volumes of polytopes

Abstract

In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers $d\ge 2$ and $n\ge d+1$ 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.