PomeLemma 17.13.10.C6092031D5B3 Let $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$. Then
\[ \sup _ X \Gamma _ t^*=t\Gamma _{\max } \]
for all $t>0$.
In particular, $t\mapsto \Gamma ^*_ t-t\Gamma _{\max }$ is a decreasing function in $t>0$.
Lemma 17.13.10C6092031D5B3
Proof. Since $\Gamma _{\tau }\leq 0$ for all $\tau <\Gamma _{\max }$, we have
\[ \sup _ X \Gamma _ t^*\leq t\Gamma _{\max } \]
for all $t>0$. Conversely, for each $\tau <\Gamma _{\max }$,
\[ \sup _ X\Gamma _ t^*\geq \sup _ X(t\tau +\Gamma _{\tau })=t\tau . \]
Letting $\tau \to \Gamma _{\max }$ gives the reverse inequality. The final assertion follows directly from the formula
\[ \Gamma _ t^*-t\Gamma _{\max }=\sup _{\tau <\Gamma _{\max }}\Bigl (\Gamma _\tau -t(\Gamma _{\max }-\tau )\Bigr ). \]
This completes the proof. $\square$
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