PomeLemma 16.28.45.F64C3174FCDB Let $q$ be bounded and upper semicontinuous on $K$, and suppose that raw contact holds for the parameter $\delta >0$ and the obstacle $q$ on $K$, that is, $V_{\delta ,q}=q$ on $K$. If $K$ is locally pluriregular at $a$, then $F_{\delta ,K,q}(a)=q(a)$ and $K$ is locally $(\delta ,q)$-regular at $a$. These equalities also hold with $\delta $ replaced by any larger parameter.
Lemma 16.28.45F64C3174FCDB
Proof. Put $M_ r=\sup _{K_ r}q$. Raw contact, monotonicity of envelopes with respect to the compact set, and (16.28.17) for $K_ r$ give
\[ q(a)\leq F_{\delta ,K,q}(a) \leq F_{\delta ,K_ r,q}(a) \leq M_ r+\delta V_{K_ r}^*(a)=M_ r. \]
Since $q$ is upper semicontinuous and $a\in K_ r$, we have $M_ r\downarrow q(a)$ as $r\downarrow 0$, which gives $F_{\delta ,K,q}(a)=q(a)$.
Every compact subset $E\subseteq K$ inherits raw contact: on $E$, the raw envelope for $E$ is at least the raw envelope for $K$, which equals $q$, and at most the obstacle $q$. For fixed $s>0$, apply the same bounds with $r<s$ and $K_ s$ in place of $K$, and let $r\downarrow 0$. This proves local $(\delta ,q)$-regularity. By theorem 16.28.14, raw contact also holds for every larger parameter, and applying what has been proved to each of them gives the last assertion. $\square$
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