Laplacian polytopes of simplicial complexes

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 $(d+1)$-simplex $\mathrm{\partial }({\sigma }_{d+1})$. If $d$ is odd, then as for graphs, the $d$-th Laplacian polytope turns out to be a $(d+1)$-simplex in this case. If $d$ is even, we show that the $d$-th Laplacian polytope of $\mathrm{\partial }({\sigma }_{d+1})$ 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 $\mathrm{\partial }({\sigma }_{d+1})$. This enables us to to compute the normalized volume and to show that the $h^{\ast }$-polynomial is real-rooted and unimodal, if $d$ is odd and even, respectively.