Proposition 18.19.2.36E07E6B83DA For these data, $Q=P_{\gamma _\sigma }[u]_{\mathcal I}$. The following are equivalent:
\[ \begin{gathered} P_{\gamma _\sigma }[u]=P_{\gamma _\sigma }[u]_{\mathcal I}, \qquad \mu _ i=E_ i\quad \text{for every }i,\\ u\in \mathcal E_{\mathrm{env}}(Y,\gamma _\sigma ,Q),\\ P_{\gamma _\sigma ,F}[u]=P_{\gamma _\sigma ,F}[u]_{\mathcal I} \quad \text{for every smooth real }F\text{ on }Y. \end{gathered} \]
The obstacle in the last condition may mix the curve coordinates. Curve backgrounds of zero area are allowed, and the models may have zero volume.
Proof.
For the zero obstacle, proposition 18.16.2 identifies the two envelopes with $\sum _ iq_ i\circ \pi _ i$ and $Q$, respectively. Since the coordinates are independent and all these functions have supremum zero, the two sums are equal if and only if $q_ i=r_ i$ for every $i$. By corollary 13.29.2, this is equivalent to $\mu _ i=E_ i$, and also to $u_ i\in \mathcal E_{\mathrm{env}}(C_ i,\sigma _ i,r_ i)$, for every $i$, including backgrounds of zero area. The preceding lemma identifies these memberships with membership in the relative class on the product. Finally, proposition 18.12.2 gives equality below every smooth obstacle, and taking $F=0$ gives the reverse implication. This proves all four equivalences.
$\square$
Pome
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