Proposition 17.12.9.B7CD1B90CFDA Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$. Then the map
\[ [-\infty ,\Gamma _{\max })\rightarrow \mathbb {R},\quad \tau \mapsto \log \int _ X \theta _{\Gamma _{\tau }}^ n \]
is continuous and concave on $(-\infty ,\Gamma _{\max })$.
Proof.
The concavity of this function follows from theorem 13.13.5 and theorem 13.13.11. The continuity at $-\infty $ is a consequence of corollary 13.13.12. The continuity on $(-\infty ,\Gamma _{\max })$ follows from the fact that finite concave functions on open intervals are continuous.
$\square$
Pome
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