Polyhedral volume ratios, Izmestiev's Colin de Verdiere matrices and Spectral Gaps

TL;DR

This paper explores the relationship between polyhedral volume ratios and spectral properties of matrices related to polytopes, introducing new geometric constructions and analyzing spectral gaps in specific symmetric polytopes.

Contribution

It establishes a novel relation between volumes of dual simplices and applies it to construct Colin de Verdiere matrices, also studying spectral gaps in vertex transitive polytopes.

Findings

Relation between volumes of dual simplices and polytope properties

Construction of Colin de Verdiere matrices from geometric data

Maximal spectral gaps occur in equilateral polytopes

Abstract

We present a relation between volumes of certain lower dimensional simplices associated to a full-dimensional primal and polar dual polytope in R^k. We then discuss an application of this relation to a geometric construction of a Colin de Verdiere matrix by Ivan Izmestiev. In the second part of the paper, we introduce a variation of vertex transitive polytopes, translate their associated Colin de Verdiere matrices into random walk matrices, and investigate extremality properties of the spectral gaps of these random walk matrices in two concrete examples - permutahedra of Coxeter groups and polytopes associated to the pure rotational tetrahedral group - where maximal spectral gaps correspond to equilateral polytopes.