Pome

Proposition 16.29.3160B15BF08FE

Proposition 16.29.3.160B15BF08FE A curve $\Gamma \in \mathscr T_{a,c}(\theta )$ is an envelope test curve with initial potential $\varphi =\Gamma _ a$. Its Legendre transform

\[ U_ t=\Bigl (\sup _{\tau <c}(\Gamma _\tau +t\tau )\Bigr )^*\qquad (t>0) \]

is an envelope ray with initial potential $\varphi $ and slope $c$, in the sense of definition 17.17.3, and satisfies

\[ \varphi +at\leq U_ t\leq \varphi +ct\qquad (t>0). \]

In particular $U_ t\sim \varphi $ for every $t>0$.

Proof. By corollary 18.4.4, a $P_\theta [\cdot ]_{\mathcal I}$-fixed potential is also fixed by the ordinary envelope $P_\theta $. The regularized supremum of the curve is $\Gamma _ a$, since the curve is decreasing and constant for $\tau \leq a$. Hence all the conditions of definition 17.17.2 hold, and theorem 17.17.5 applies. The term $\tau =a$ in the transform gives the lower bound, and the inequalities $\Gamma _\tau \leq \varphi $ for $\tau <c$ give the upper bound. $\square$

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