(clearly for $\XX_\D$, the role of the dual lattices $M,N$ is reversed with respect to $\P_\D$). This is a direct consequence of the following

\[prop:Fmatricedipolitopo\] Given an identification $M\cong\Z^n$ and a lattice polytope $$\D:=\operatorname{Conv}(\{\m_i\in M\,|\,i=1,\ldots,m\})$$ let $V_\D=\left(
           \begin{array}{ccc}
             \v_1 & \cdots & \v_m \\
           \end{array}
         \right)$ be the $n\times m$ integer matrix defined by the generators $\v_i$ of the semi-groups $\langle\m_i\rangle\cap M$, for any $1\leq i\leq m$. Then $V_\D$ is a reduced $F$-matrix if and only if $\0\in\operatorname{Int}\D$. In particular, if $\0\in\operatorname{Int}\D$ then $V_\D$ is a fan matrix of $\XX_\D$.

Assume $V_\D$ is an $F$-matrix. Then $V_\D$ is clearly reduced as all the $\v_i$’s are primitive. Moreover, choosing the first column $\v_1$ of $V_\D$, the opposite vector $-\v_1$ belongs to the cone $\langle\v_2,\ldots,\v_n\rangle$, by . Then $$\0=\v_1-\v_1=\v_1 +\sum_{j=2}^m \lambda_j\v_j=\sum_{i=1}^m \mu_i\m_i$$ where $$\mu_1={\|\v_1\|\over\|\m_1\|}>0\ ,\quad \forall\,j\geq 2\quad \mu_j=\lambda_j{\|\v_j\|\over\|\m_j\|}>0$$ One can then conclude that $\0\in\operatorname{Int}\operatorname{Conv}(\{\m_k\}_{k=1}^m)=\operatorname{Int}\D$ by setting $\mu:=\sum_k\mu_k$ and writing $\0=\sum_k (\mu_k/\mu)\,\m_k$.

Viceversa, assume $\0\in\operatorname{Int}\D=\operatorname{Int}\operatorname{Conv}(\{\m_k\}_{k=1}^m)$. Conditions (a), (c) and (d) in Definition \[def:Fmatrice\] are clearly satisfied. To show that $V_\D$ is $F$-complete, for any vector $\v\in M_\R$ consider the polytope $$\D':=\operatorname{Conv}(\m_1\,\ldots,\m_m,-\v)\subseteq M_\R$$ Since $\D\subseteq\D'$, one has that $\0\in\operatorname{Int}\D'$, meaning that $$\exists\,\mu_1>0,\ldots,\mu_m>0,\mu>0\,:\quad \sum_k\mu_k +\mu=1\ ,\quad\0=\sum_k\mu_k\m_k-\mu\v$$ meaning that $$\v=\sum_k\l_k\v_k\ ,\quad\text{with}\quad \forall\,k\ \l_k={\mu_k\,\|\m_k\|\over\mu\,\|\v_k\|}>0\ \Longrightarrow\ \v\in\langle V_\D\rangle$$

