We call $\Pi$ a dual $\Delta$-complex if it corresponds to the convex polyhedron $\Delta\subset\mathbb{R}^{n+1}$ by Proposition 1.4. We call $\Pi$ a maximal polyhedral complex if the elements of the corresponding subdivision from Corollary 1.5 are simplices of volume $\frac{1}{(n+1)!}$ (a so-called unimodular lattice triangulation).