Let $e_{i}$, $1\leq i\leq n$, and $f_{j}$, $1\leq j\leq r$, be the $i$-th and $(n+j)$-th vectors of the standard basis of $N=\mathbb{Z}^{n+r}$. Set $f_{0}=-f_{1}-\cdots-f_{r}$ and $e_{0}=a_{0}f_{0}+\cdots+a_{r}f_{r}-e_{1}-\cdots-e_{n}$. The fan $\Sigma$ corresponding to $\mathbb{P}(E)$ is the fan in $N_{\mathbb{R}}$ whose maximal cones are the convex hull of the rays generated by the vectors

$$e_{0},\cdots,e_{k-1},e_{k+1},\cdots,e_{n},f_{0},\cdots,f_{\ell-1},f_{\ell+1},\cdots,f_{r}$$

for $0\leq k\leq n,0\leq\ell\leq r$. This is a complete regular fan.

The support function $\Psi\colon N_{\mathbb{R}}\to{\mathbb{R}}$ corresponding to the universal line bundle ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ is defined, for $u\in\mathbb{R}^{n}$ and $v\in\mathbb{R}^{r}$, as

$$\Psi(u,v)=\mathop{\min_{0\leq k\leq n}}_{0\leq\ell\leq r}(a_{\ell}u_{k}+v_{\ell}),$$

where, for short, we have set $u_{0}=v_{0}=0$.

The polytope $\Delta$ in $M_{\mathbb{R}}=\mathbb{R}^{n}\times\mathbb{R}^{r}$ associated to $(\Sigma,\Psi)$ is

$$\Big\{(x,y)|y_{1},\dots,y_{r}\geq 0,\ \sum_{\ell=1}^{r}y_{\ell}\leq 1,\ x_{1},\dots,x_{n}\geq 0,\ \sum_{k=1}^{n}x_{k}\leq L(y)\Big\}$$

with $L(y)=a_{0}+\sum_{\ell=1}^{r}(a_{\ell}-a_{0})y_{\ell}$. Using the convention $y_{0}=1-\sum_{\ell=1}^{r}y_{\ell}$ and $x_{0}=L(y)-\sum_{k=1}^{n}x_{k}$, then $L(y)=\sum_{\ell=0}^{r}a_{\ell}y_{\ell}$ and the polytope $\Delta$ can be written as

$$\Big\{(x,y)|y_{0},\dots,y_{r}\geq 0,\ x_{0},\dots,x_{n}\geq 0\Big\}.$$

The Legendre-Fenchel dual of $\psi_{\infty}$ is the concave function $\psi_{\infty}^{\vee}\colon\Delta\to\mathbb{R}$ defined, for $(x,y)\in\Delta$, as

$$\psi^{\vee}_{\infty}(x,y)=-\frac{1}{2}\left(\varepsilon_{r}(y_{1},\dots,y_{r})+L(y)\cdot\varepsilon_{n}\left(\frac{x_{1}}{L(y)},\dots,\frac{x_{n}}{L(y)}\right)\right),$$

where, for $k\geq 0$, $\varepsilon_{k}$ is the function defined in (3.54). For $v\neq\infty$, the concave function $\psi^{\vee}_{v}$ is the indicator function of $\Delta$.