Let $\Sigma$ be a complete fan on $N_{\mathbb{R}}$. Let ${\overline{L}}=(L,\|\cdot\|)$ be a toric line bundle on $X_{\Sigma}$, generated by global sections, and equipped with an approachable toric metric. Choose any toric section $s$ of $L$; let $\Psi$ be the associated support function on $\Sigma$, and put $\Delta_{\Psi}=\operatorname{stab}(\Psi)$ for the associated polytope. Then, the toric local height of $X_{\Sigma}$ with respect to ${\overline{L}}$ is given by

(6.7)

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=(n+1)!\lambda_{K}\int_{\Delta_{\Psi}}\psi_{{\overline{L}},s}^{\vee}\,\text{\rm d}\operatorname{vol}_{M}.$$

where $\,\text{\rm d}\operatorname{vol}_{M}$ is the unique Haar measure of $M_{\mathbb{R}}$ such that the co-volume of $M$ is one and $\psi_{{\overline{L}},s}^{\vee}$ is the Legendre-Fenchel dual to the function $\psi_{{\overline{L}},s}$ associated to $({\overline{L}},s)$ in Definition 5.14.