A polyhedral complex ${\mathscr{C}}$ in $N_{\mathbb{R}}$ is a finite set of polyhedra with the following two properties: Every polyhedron in ${\mathscr{C}}$ has all its closed faces in ${\mathscr{C}}$. If $\Delta,\sigma\in{{\mathscr{C}}}$, then $\Delta\cap\sigma$ is a closed face of $\Delta$ and $\sigma$. Note here that the empty set and also $\sigma$ are allowed as closed faces of a polyhedron $\sigma$ (see [Gu12], Appendix A, for details).

A polyhedral complex ${\mathscr{C}}$ is called integral $G$-affine for a subgroup $G$ of ${\mathbb{R}}$ if every polyhedron of ${\mathscr{C}}$ is integral $G$-affine. The support $|{\mathscr{C}}|$ of ${\mathscr{C}}$ is the union of all polyhedra in ${\mathscr{C}}$. The polyhedral complex ${\mathscr{C}}$ is called pure dimensional of dimension $n$ if every maximal polyhedron in ${\mathscr{C}}$ has dimension $n$. We will often use the notation ${\mathscr{C}}_{k}:=\{\sigma\in{\mathscr{C}}\mid\dim(\sigma)=k\}$ for $k\in{\mathbb{N}}$.