In dimension $1$, a polytope is an interval of the form $\Delta=[m_{0},m_{1}]$ for some $m_{i}\in\mathbb{Z}$. The corresponding roof function in (7.20) writes down, for $x\in[m_{0},m_{1}]$, as

(7.31)

$$\vartheta(x)=-\sum_{i=1}^{r}c_{i}\ell_{i}(x)\log(\ell_{i}(x))$$

for affine function $\ell_{i}=u_{i}x-\lambda_{i}$ which take non negative values on the $\Delta$ and $c_{i}>0$

The polarized toric variety corresponding to $\Delta$ is $\mathbb{P}^{1}$ together with the ample divisor $m_{1}[(0:1)]-m_{0}[(1:0)]$. Write $L={\mathcal{O}}_{\mathbb{P}^{1}}(x_{1}-x_{0})$ for the associate line bundle and ${\overline{L}}$ for the adelic metrized line bundle corresponding to the function $\vartheta$. The Legendre-Fenchel dual to $-c_{i}\ell_{i}(x)\log(\ell_{i}(x))$ is the function $f_{i}\colon\mathbb{R}\rightarrow\mathbb{R}$ defined, for $v\in\mathbb{R}$, by

$$f_{i}(v)=\frac{\lambda_{i}}{u_{i}}v-c_{i}{\operatorname{e}}^{-1-\frac{v}{{c_{i}u_{i}}}}.$$

Therefore, the function $\psi=\vartheta^{\vee}$ is the sup-convolution of these function, namely $\psi=f_{1}\boxplus\dots\boxplus f_{m}$ For the height, a simple computation shows that

$${\operatorname{h}_{\overline{L}}(\mathbb{P}^{1})}=\int_{m_{0}}^{m_{1}}\vartheta\,\text{\rm d}x=\sum_{i=1}^{r}\frac{c_{i}}{4u_{i}}\Big[\ell_{i}(x)^{2}\left(1-2\log(\ell_{i}(x))\right)\Big]^{m_{1}}_{m_{0}}$$