Let $A\subset\mathbb{Z}^{n+1}$ be a finite set and let $v:A\to\mathbb{R}$ be any function. Let $\Delta\subset\mathbb{R}^{n+1}$ be the convex hull of $A$. We associate the following polyhedral complex $\Pi_{v}$ to $v$.

Take the Legendre transform $L_{v}:\mathbb{R}^{n+1}\to\mathbb{R}$ of $v$

Here $x,y\in\mathbb{R}^{n+1}$ and $xy$ is their scalar product. Since the maximum is taken over a finite set, the result $L_{v}$ is a convex piecewise-linear function. We define $\Pi_{v}$ as the corner locus of $L_{v}$ (recall that this is the set of points where $L_{v}$ is not smooth).

To present Example 1 as a special case of Example 2 we take the vertices of the standard simplex

for $A$ and set $v\equiv 0$.