Let $\Sigma$ be a complete fan in $N$ and $\Psi$ a support function on $\Sigma$. Write $L=L_{\Psi}$ and $s=s_{\Psi}$. Let $\psi\colon N_{\mathbb{R}}\to\mathbb{R}$ a piecewise affine concave function with $\operatorname{rec}(\psi)=\Psi$, that has an $H$-representation

$$\psi=\min_{i=0,\dots,r}\{m_{i}+l_{i}\},$$

with $m_{i}\in M_{\mathbb{Q}}$ and $l_{i}\in\mathbb{R}$ in the Archimedean case and $l_{i}\in\mathbb{Q}$ in the non-Archimedean case. Then there is an equivariant morphism $\varphi\colon X_{\Sigma}\to\mathbb{P}^{r}$, an integer $e>0$ and an isomorphism $L^{\otimes e}\simeq\varphi^{\ast}\mathcal{O}(1)$ such that the metric induced on $L^{{\text{\rm an}}}$ by the canonical metric of $\mathcal{O}(1)^{{\text{\rm an}}}$ agrees with $\|\cdot\|_{\psi}$.