If $\Psi$ is a strictly concave function on $\Sigma$, then $\Delta_{\Psi}$ is an $n$-dimensional lattice polytope. Conversely, if $\Delta$ is a lattice polytope in $M_{\mathbb{R}}$, then $\Psi_{\Delta}$, the support function of $\Delta$, is a strictly concave function on the complete fan $\Sigma_{\Delta}=\Pi(\Psi_{\Delta})$ (see examples 3.71 and 3.76). Therefore, the result follows from Theorem 4.18 and the construction of Remark 4.40. ∎