Let $\Pi\subset\mathbb{R}^{n+1}$ be an $n$-dimensional balanced polyhedral complex. By Proposition 1.4 there is a convex lattice polyhedron $\Delta$ dual to $\Pi$. We define $\bar{\Pi}\subset\Delta$, the compactification of $\Pi$, by taking the closure of $\Phi_{\Delta}(\Pi)$ in $\Delta$. We call $\bar{\Pi}\smallsetminus\Phi_{\Delta}(\Pi)$ the boundary of $\bar{\Pi}$. For convenience from now on we identify $\Pi$ and $\Phi_{\Delta}(\Pi)$.