We continue with Example 5.26. Let $\mathbb{Z}^{r}$ be the standard lattice of rank $r$, $\Delta^{r}$ the standard simplex of dimension $r$ and $\Sigma_{\Delta^{r}}$ the fan of $\mathbb{R}^{r}$ associated to $\Delta^{r}$. The corresponding toric variety is $\mathbb{P}^{r}$. Let $H\colon N\to\mathbb{Z}^{r}$ be an injective linear morphism such that $H(N)$ is a saturated sublattice. Denote $m_{i}=e_{i}^{\vee}\circ H\in M$, $i=1,\dots,r$. Let $\Sigma$ the regular fan on $N$ defined by $H$ and $\Sigma_{\Delta^{r}}$. Let $\Psi_{\Delta^{r}}$ be the support function of $\Delta^{r}$ and let $\Psi=\Psi_{\Delta^{r}}\circ H$. Explicitly,

$$\Psi(v)=\min(0,m_{1}(v),\dots,m_{r}(v)).$$

Let $p\in\mathbb{P}^{r}_{0}(K)$ and $u={\operatorname{val}}_{K}(p)\in\mathbb{R}^{r}$. Write $u=(u_{1},\dots,u_{r})$. If $p=(1:\alpha_{1}:\dots:\alpha_{r})$, then $u_{i}=\frac{-\log(|\alpha_{i}|)}{\lambda_{K}}$. There is an equivariant morphism $\varphi:=\varphi_{p,H}\colon X_{\Sigma}\to\mathbb{P}^{r}$. Consider the toric line bundle with toric section determined by $\Psi_{\Delta^{r}}$ with the canonical metric and denote by $({\overline{L}},s)$ the induced toric line bundle with toric section on $X_{\Sigma}$ equipped with the induced metric. Then

$$\psi_{{\overline{L}},s}(v)=\min(0,m_{1}(v)+u_{1},\dots,m_{r}(v)+u_{r}).$$

Thus $\Delta=\operatorname{stab}(\psi_{{\overline{L}},s})=\operatorname{conv}(0,m_{1},\dots,m_{r})=H^{\vee}(\Delta^{r})$. By Proposition 3.64 the Legendre-Fenchel dual $\psi_{{\overline{L}},s}^{\vee}\colon\Delta\to\mathbb{R}$ is given by

$$\psi_{{\overline{L}},s}^{\vee}(x)=\sup\bigg\{\sum_{j=1}^{r}-\lambda_{j}u_{j}\bigg|\ \lambda_{j}\geq 0,\sum_{j=1}^{r}\lambda_{j}\leq 1,\ \sum_{j=1}^{r}\lambda_{j}a_{j}=x\bigg\}\ \text{ for }x\in\Delta.$$

This function is the upper envelope of the extended polytope of $M_{\mathbb{R}}\times\mathbb{R}$,

$$\operatorname{conv}\left((0,0),(m_{1},-u_{1}),\dots,(m_{r},-u_{r})\right),$$

Similarly, the roof function $\vartheta_{{\overline{L}},s}=\lambda_{K}\psi_{{\overline{L}},s}^{\vee}$ is the upper envelope of the extended polytope

$$\operatorname{conv}\left((0,0),(m_{1},\log|\alpha_{1}|),\dots,(m_{r},\log|\alpha_{r}|)\right).$$