Let $\Pi$ be a complete SCR complex in $N_{\mathbb{R}}$ and ${\mathcal{X}}_{\Pi}$ its associate toric scheme over $S$. Let $\psi$ be an H-lattice function on $\Pi$ and $D_{\psi}$ the corresponding $\mathbb{T}$-Cartier divisor on ${\mathcal{X}}_{\Pi}$.

1. (1)

   The following properties are equivalent:

   1. (a)

      $D_{\psi}$ is ample;

   2. (b)

      $D_{\psi}\cdot C>0$ for every vertical curve $C$ contained in $X_{\Pi,o}$;

   3. (c)

      $D_{\psi}\cdot V(\Lambda)>0$ for every $(n-1)$-dimensional polyhedron $\Lambda\in\Pi$;

   4. (d)

      The function $\psi$ is strictly concave on $\Pi$.

2. (2)

   The following properties are equivalent:

   1. (a)

      $D_{\psi}$ is generated by global sections;

   2. (b)

      $D_{\psi}\cdot C\geq 0$ for every vertical curve $C$ contained in $X_{\Sigma,o}$;

   3. (c)

      $D_{\psi}\cdot V(\Lambda)\geq 0$ for every $(n-1)$-dimensional polyhedron $\Lambda\in\Pi$;

   4. (d)

      The function $\psi$ is concave.