(\[ss-intersezione\]) then $$\label{polare2}
  \D^*=\operatorname{Conv}(\{a_\Phi^{-1}\n_\Phi\,|\,\forall\,\Phi<^1\D\})\subseteq N_\R$$ (see [@CLS Exer. 2.2.1]). Clearly, in general, $\D^*$ is not a lattice polytope in $N$, even if $\D$ is a lattice polytope in $M$. A lattice polytope $\D$ is called *reflexive* if $\0\in\operatorname{Int}\D$ and $\D^*$ is still a lattice polytope. By [@Batyrev94 Thm. 4.1.6] $$\D\ \text{is reflexive}\ \Longleftrightarrow\ \operatorname{Int}\D\cap M=\{\0\}$$ Given a divisor $D=\sum_{\rho\in\Si(1)}a_\rho D_\rho\in\Weil(X(\Si))$, the following polyhedron $$\label{div-politopo}
  \D_D:=\{\m\in M_\R\,|\,\forall\,\rho\in\Si(1)\quad\langle\m,\v_\rho\rangle\geq -a_\rho\}=\{\m\in M_\R\,|\,V^T\cdot\m \geq -\aa\}$$ is called the *polyhedron associated to $D$*, where $V=\left(\v_\rho\right)_{\rho\in\Si(1)}$ is a fan matrix of $X$ and $\aa=\left(a_\rho\right)_{\rho\in\Si(1)}$ is the column vector of coefficients of $D$. In general it is not a polytope, but just a polyhedron as intersection of a finitely many closed half spaces.

\[prop:gg\] If $X(\Si)$ is complete then, for any $D$ in $\Weil(X)$, the associated polyhedron $\D_D$ is a polytope. Moreover:

1.  $D$ is *basepoint free*, that is $\cO_X(D)$ is generated by global section, if and only if $\D_D=\operatorname{Conv}(\{\m_\s\in M\,|\,\s\in\Si(n)\})$,

2.  $D$ is ample if and only if $\D_D=\operatorname{Conv}(\{\m_\s\in M\,|\,\s\in\Si(n)\})$ and $\s\neq\s'$ implies $\m_\s\neq\m_{\s'}$.

Recall that a Weil divisor $D$ is *semi-ample* if a positive multiple $kD$, $k\in\N$, is basepoint free (hence Cartier). In particular, if $X(\Si)$ is complete and $D$ semi-ample, then $$\D_{kD}=k\D_D=\operatorname{Conv}(\{\m_\s\in M\,|\,\s\in\Si(n)\})$$ is a 