Let notation be as in Definition 7.23. Then $\operatorname{h}_{\overline{L}}(X_{\Sigma_{\Delta}})$ equals

$${(n+1)!}\sum_{i=1}^{r}c_{i}\sum_{V\in\Delta(u_{i})}\sum_{k=0}^{\dim(V)}C_{k}(\Delta,u_{i},V)\frac{\ell_{i}(V)^{n-k+1}}{(n-k+1)!}\left(\sum_{j=2}^{n-k+1}\frac{1}{j}-\log(\ell_{i}(V))\right).$$

Suppose furthermore that $\Delta\subset\mathbb{R}^{n}$ is a simplex, $r=n+1$ and that $\ell_{i}$, $i=1,\dots,n+1$, are affine functions such that $\Delta=\bigcap_{i}\{x\in M_{\mathbb{R}}|\ell_{i}(x)\geq 0\}$. Then

(7.28)

$$\operatorname{h}_{\overline{L}}(X_{\Sigma_{\Delta}})=n!\operatorname{vol}_{M}(\Delta)\sum_{i=1}^{n+1}c_{i}\ell_{i}(\nu_{i})\bigg(\sum_{j=2}^{n+1}\frac{1}{j}-\log(\ell_{i}(\nu_{i}))\bigg).$$

where $\nu_{i}$ is the unique vertex of $\Delta$ not contained in the facet defined by $\ell_{i}$.