Example 7.35 . [02XX]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Example 7.35.

The Fubini-Study metric of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ corresponds to the case when $\Gamma$ and $\Delta$ are the standard simplex $\Delta^{n}$ and $c=1/2$ . In that case, the average entropy of the random variable $\beta_{x}$ is

\frac{1}{n!}\int_{\Delta^{n}}{\mathcal{E}}(x)\,\text{\rm d}\operatorname{vol}_{n}=\frac{2\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})}{(n+1)}=\sum_{j=2}^{n+1}\frac{1}{j}.