Proposition 7.34 . [02XV]

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Proposition 7.34.

With the above notation,

\frac{1}{\operatorname{vol}_{n}(\Delta)}\int_{\Delta}{\mathcal{E}}(x)\,\text{\rm d}\operatorname{vol}_{n}=\frac{1}{n!\operatorname{vol}_{n}(\Gamma)}\bigg(\frac{\operatorname{h}_{{\overline{L}}}(X)}{c(n+1)\deg_{L}(X)}-{\log(n!\operatorname{vol}_{n}(\Gamma))}\Big(\sum_{F}\lambda_{F}\Big)\bigg)

where the sum is over the facets $F$ of $\Gamma$ . In particular, if $\Gamma=\Delta$ ,

\frac{1}{\operatorname{vol}_{n}(\Delta)}\int_{\Delta}{\mathcal{E}}(x)\,\text{\rm d}\operatorname{vol}_{n}=\frac{\operatorname{h}_{{\overline{L}}}(X)}{c(n+1)\deg_{L}(X)^{2}}-\frac{\log(\deg_{L}(X))}{\deg_{L}(X)}\Big(\sum_{F}\lambda_{F}\Big).