PomeCorollary 18.22.4.DFA6018C6C86 Let $U=v\circ p$ and $R=P_\theta [U]_{\mathcal I}$. The following are equivalent: $P_\theta [U]=P_\theta [U]_{\mathcal I}$; $\mu _ v$ is atomic; $U\in \mathcal E_{\mathrm{env}}(X,\theta ,R)$; and the ordinary and ideal envelopes of $U$ agree below every smooth real obstacle on $X$. At zero area all four conditions hold.
Corollary 18.22.4DFA6018C6C86
Proof. Put $r=P_\beta [v]_{\mathcal I}$. The envelope bijection gives $R=r\circ p$. By corollary 13.29.2, base envelope equality is equivalent to atomic polar curvature and to $v\in \mathcal E_{\mathrm{env}}(C,\beta ,r)$. For each stretching parameter $b\geq 1$, descent and surjectivity identify competitors satisfying $b^{-1}h+(1-b^{-1})r\leq v$ with their upstairs counterparts. This uses only convex combinations and remains valid at poles. Thus the relative-class conditions correspond. Apply proposition 18.12.2 for equality below every smooth obstacle on $X$; the converse includes the zero obstacle. The zero-area assertion follows from the smooth singleton. $\square$
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