Note that several different functions $v$ define the same complex $\Pi_{v}$ by the construction of Example 2. Here is the list of ambiguities.

1. (1)

   Let $v^{\prime}=v+\operatorname{const}:A\to\mathbb{R}$ be a function different with $v$ by a constant. Then $\Pi_{v}=\Pi_{v^{\prime}}$.

2. (2)

   Let $A^{\prime}=A+c$, where $c\in\mathbb{Z}^{n+1}$ and $v^{\prime}:A^{\prime}\to\mathbb{R}$ is defined by $v^{\prime}(z+c)=v(z)$. Then $\Pi_{v}=\Pi_{v^{\prime}}$.

3. (3)

   Let $A^{\prime}$ be such that its convex hull $\Delta^{\prime}$ coincides with $\Delta$, the convex hull of $A$. Let $\underline{v}$ (resp. $\underline{v^{\prime}}$) be the maximal convex function such that $\underline{v}\leq v$ (resp. $\underline{v^{\prime}}\leq v^{\prime}$). Suppose that $\underline{v}=\underline{v^{\prime}}$. Then $\Pi_{v}=\Pi_{v^{\prime}}$.