First we define a convex piecewise-linear function $H$ whose corner locus is $\Pi$ and then choose a function $v$ such that $H$ is the Legendre transform $L_{v}$ of $v$. Note that the finiteness condition in Definition 1 implies that there are finitely many connected components in $\mathbb{R}^{n+1}\smallsetminus\Pi$.

We define the function $H$ inductively. Choose any connected component $D_{0}$ of $\mathbb{R}^{n+1}\smallsetminus\Pi$ as a “reference component”. Define $H|_{D_{0}}\equiv 0$. Suppose that $D^{\prime}$ is a component of $\mathbb{R}^{n+1}\smallsetminus\Pi$ such that there exists an adjacent component $D$ where $H$ is already defined.

Let $F$ be the $n$-cell of of $\Pi$ separating $D$ from $D^{\prime}$. Let $c_{F}$ be the covector associated to $F$ (recall that the weight of $F$ is incorporated into $c_{F}$) with the co-orientation directed from $D$ to $D^{\prime}$. Let $l_{D}:\mathbb{R}^{n+1}\to\mathbb{R}$ be the linear function extending $H|_{D}$. We define $H|_{D^{\prime}}=l_{D}+c_{F}$. By the balancing condition the result does not depend on the choice of the adjacent component $D$ where $H$ is already defined.

To define $v$ we take the Legendre transform of $H$. This amounts to associating each component $D$ a point $z\in\mathbb{Z}^{n+1}$ equal to the gradient of $H|_{D}$ and setting $v(z)=l_{D}(0)$. Thus, the number of elements of the set $A$ is equal to the number of components of $\mathbb{R}^{n+1}\smallsetminus\Pi$.

The ambiguity Remark 1.3.3 comes from taking the Legendre transform of non-convex functions $v$. It coincides with the Legendre transform of the underlying convex function $\underline{v}$. (In fact, nothing changes if we assume that $v$ is defined on the whole $\mathbb{Z}^{n+1}$ by letting $v(z)=+\infty$ for $z\notin A$.) The ambiguities Remark 1.3.1 and 1.3.2 come from the ambiguity in assigning a linear function for $H|_{D_{0}}$. ∎