Suppose that $\Pi\subset\mathbb{R}^{n+1}$ is a weighted balanced proper rational polyhedral complex. Then there exists a finite set $A\subset\mathbb{Z}^{n+1}$ and a function $v:A\to\mathbb{Z}$ such that $\Pi=\Pi_{v}$ (see Example 2). The convex hull $\Delta\subset\mathbb{R}^{n+1}$ of $A$ is unique up to a translation in $\mathbb{Z}^{n+1}$. The choice of the function $v$ is unique up to the ambiguity of Remark 1.3.