Laplacian polytopes of simplicial complexes

Abstract

Given a (finite) simplicial complex, we define its i-th Laplacian polytope as the convex hull of the columns of its i-th Laplacian matrix. This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer. After studying basic properties of these polytopes, we focus on the d-th Laplacian polytope of the boundary of a pd `1q-simplex Bpσ d`1 q. If d is odd, then as for graphs, the d-th Laplacian polytope turns out to be a pd `1qsimplex in this case. If d is even, we show that the d-th Laplacian polytope of Bpσ d`1 q is combinatorially equivalent to a d-dimensional cyclic polytope on d `2 vertices. Moreover, we provide an explicit regular unimodular triangulation for the d-th Laplacian polytope of Bpσ d`1 q. This enables us to to compute the normalized volume and to show that the h ˚-polynomial is real-rooted and unimodal, if d is odd and even, respectively.