We start by associating to $v$ a certain lattice subdivision ${\mathcal{D}}_{v}$ of $\Delta$. Let $O\Gamma(v)$ be the overgraph of $v$, i.e. the set of vertical rays upwards in $\mathbb{R}^{n+1}\times\mathbb{R}$ starting at the points of the graph of $v$. The convex hull of $O\Gamma(v)$ is a semi-infinite closed polyhedral domain. The projections of its finite faces to $\mathbb{R}^{n+1}$ form the subdivision ${\mathcal{D}}_{v}$.

We claim that $\Pi_{v}$ is a polyhedral complex dual to ${\mathcal{D}}_{v}$. Namely, a $k$-dimensional polyhedron $\Delta^{\prime}$ in ${\mathcal{D}}_{v}$, $k>0$, gives a $(n+1-k)$-cell of $\Pi_{v}$. This cell is compact iff $\Delta^{\prime}\subset\Delta$.

This claim follows from the duality property of the Legendre transform. Consider the function $\tilde{v}$ whose graph is is given by the lower boundary of the convex hull of $O\Gamma(v)$. If $v$ is convex then the function $\tilde{v}$ extends $v$ and is defined on the whole polyhedron $\Delta$, not just on its lattice points. It is a convex piecewise-linear function. The Legendre transform of $v$ coincides with the Legendre transform of $\tilde{v}$. (In fact the function $\tilde{v}$ can be defined by applying the Legendre transform to $v$ twice.) By duality, the graph of $L_{\tilde{v}}$ has the facets en lieu of the vertices of the graph of $\tilde{v}$ and so on. ∎