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

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.