Lemma 8.17 . [02YF]

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Lemma 8.17.

The function $\psi_{\infty}$ is defined, for $u\in\mathbb{R}^{n}$ and $v\in\mathbb{R}^{r}$ , as

\psi_{\infty}(u,v)=-\frac{1}{2}\log\left(\sum_{j=0}^{r}{\operatorname{e}}^{-2v_{j}}\left(\sum_{i=0}^{n}{\operatorname{e}}^{-2u_{i}}\right)^{a_{j}}\right),

with the convention $u_{0}=v_{0}=0$ . It is a strictly concave function.