An integral affine structure on $Y$ is polyhedral if there is an $n$-dimensional polyhedral complex $P$, a collection of disjoint open sets $\{U_{\alpha}\}$, whose closures cover $Y$, i.e. $Y=\bigcup\overline{U}_{\alpha}$, and a continuous map $\phi:Y\to P$, which provides an affine homeomorphism of each $U_{\alpha}$ with the interior of some $n$-dimensional face of $P$. We say that the pair $(\{U_{\alpha}\},P)$ realizes the polyhedral affine structure if $P$ is minimal, which, in particular, means that there is a bijection between open sets $\{U_{\alpha}\}$ and $n$-dimensional cells of $P$.