Proposition 17.13.51F48BF7602CA

Proof. Fix $x\in X$. We want to establish (17.13.6) at $x$. We distinguish two cases. First suppose that $\Gamma _{\tau }(x)=-\infty $ for all $\tau <\Gamma _{\max }$ and hence all $\tau \in \mathbb {R}$. In this case, we have $\Gamma _ t^*(x)=-\infty $ for all $t>0$. Therefore, (17.13.6) follows trivially.

Otherwise, by 17.13.4, we know that $\Gamma ^*_ t(x)=\infty $ for all $t<0$. The relative interior of the domain of $t\mapsto \Gamma ^*_ t(x)$ is contained in $(0,\infty )$. Therefore, (17.13.6) follows from theorem 21.11.24, proposition 17.12.16. $\square$