  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
$\partial(\sigma_{d+1})$. 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 $\partial(\sigma_{d+1})$ 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 $\partial(\sigma_{d+1})$. 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.

Combinatorics