Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $X_{\Sigma}$ the corresponding toric variety. Recall the description of the projective space $\mathbb{P}^{r}$ as a toric variety given in Example 4.3. Let $H\colon N\to\mathbb{Z}^{r}$ be a linear map such that, for each $\sigma\in\Sigma$ there exist $\tau\in\Sigma_{\Delta^{r}}$ with $H(\sigma)\subset\tau$. Let $p\in\mathbb{P}^{r}_{0}(K)$. Then we have an equivariant morphism $\varphi_{p,H}\colon X_{\Sigma}\to\mathbb{P}^{r}$. Consider the support function $\Psi_{\Delta^{r}}$ on $\Sigma_{\Delta^{r}}$. Then $L_{\Psi_{\Delta^{r}}}=\mathcal{O}_{\mathbb{P}^{r}}(1)$. Write $L=\varphi^{\ast}_{p,H}L_{\Psi_{\Delta^{r}}}$, $s=\varphi^{\ast}_{p,H}s_{\Psi_{\Delta^{r}}}$ and $\Psi=H^{\ast}\Psi_{\Delta^{r}}$. Thus $(L,s)=(L_{\Psi},s_{\Psi})$.

Set $A=H+{\operatorname{val}}_{K}(p)$ for the affine map. Let $\|\cdot\|$ be the metric on $L^{{\text{\rm an}}}$ induced by the canonical metric of $\mathcal{O}(D_{\Psi_{\Delta^{r}}})^{{\text{\rm an}}}$ and let $\psi$ be the function associated to it by Proposition 5.16. By Proposition 5.24, $\psi=A^{\ast}\Psi_{\Delta^{r}}$. This is a piecewise affine concave function on $N_{\mathbb{R}}$ with $\operatorname{rec}(\psi)=\Psi$ that can be made explicit as follows.

Let $\{e_{1},\dots,e_{r}\}$ be the standard basis of $\mathbb{Z}^{r}$ and let $\{e_{1}^{\vee},\dots,e_{r}^{\vee}\}$ be the dual basis. Write $m_{i}=e_{i}^{\vee}\circ H\in M$ and $l_{i}=e_{i}^{\vee}({\operatorname{val}}_{K}(p))\in\mathbb{R}$. Then