Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi$ a virtual support function on $\Sigma$. Let $({\mathcal{X}},D,e)$ be a toric model of $(X_{\Sigma},D_{\Psi})$. Then, by Theorem 4.81, there exists a complete SCR polyhedral complex $\Pi$ in $N_{\mathbb{R}}$ with $\operatorname{rec}(\Pi)=\Sigma$ and a rational piecewise affine function $\psi$ on $\Pi$ such that $e\psi$ is an H-lattice function, $\operatorname{rec}(\psi)=\Psi$ and $({\mathcal{X}},D,e)=({\mathcal{X}}_{\Pi},D_{e\psi},e)$. Moreover, if $({\mathcal{X}}^{\prime},D^{\prime},e^{\prime})$ is another toric model that gives the function $\psi^{\prime}$, then both models are equivalent if and only if $\psi=\psi^{\prime}$. Thus, to every toric model we have associated a rational piecewise affine function $\psi$ on $\Pi$ such that $\operatorname{rec}(\psi)=\Psi$. Two equivalent models give rise to the same function.

The converse is not true. Given a rational piecewise affine function $\psi$, with $\operatorname{rec}(\psi)=\Psi$, we can find a complete SCR polyhedral complex $\Pi$ such that $\psi$ is piecewise affine on $\Pi$. But, in general $\operatorname{rec}(\Pi)$ does not agree with $\Sigma$. What we can expect is that $\Sigma^{\prime}:=\operatorname{rec}(\Pi)$ is a refinement of $\Sigma$. Therefore the function $\psi$ gives us an equivalence class of toric models of $(X_{\Sigma^{\prime}},D_{\Psi})$. But $\psi$ may not determine an equivalence class of toric models of $(X_{\Sigma},D_{\Psi})$. In Corollary 5.43 in next section we will give a necessary condition for a function $\psi$ to define an equivalence class of toric models of $(X_{\Sigma},D_{\Psi})$ and in Example 5.44 we will exhibit a function that does not satisfy this necessary condition. By contrast, as we will see in Theorem 4.97, the concave case is much more transparent.