PomeProposition 16.28.64.46B23EB58AAB If $u_1,\ldots ,u_ N\in \mathcal E_\varphi $, $N\geq 1$, then the rooftop envelope $P_\theta (\min _ iu_ i)$ is not identically $-\infty $ and belongs to $\mathcal E_\varphi $.
Proposition 16.28.6446B23EB58AAB
Proof. Adding a common constant to all $u_ i$, we may assume $u_ i\leq \varphi $. By lemma 16.28.63, proposition 16.28.62,
\[ h=\frac1N\sum _{i=1}^ N Q_ N(u_ i,\varphi )\in \mathcal E_\varphi . \]
Each summand is at most $\varphi $. For every $i$, the defining inequality of $Q_ N(u_ i,\varphi )$ therefore gives
\[ h\leq N^{-1}Q_ N(u_ i,\varphi )+(1-N^{-1})\varphi \leq u_ i. \]
These inequalities between convex combinations also hold at common poles. Thus $h$ is a competitor for $p=P_\theta (\min _ i u_ i)$, so $p$ is not identically $-\infty $, and $h\leq p\leq u_1\preceq \varphi $. Stability under increase gives $p\in \mathcal E_\varphi $, and undoing the translation preserves this conclusion. $\square$
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