Siciak extremal function

Remarks

The value is at least $0$, since $u = 0$ is admissible. For compact $E$, Siciak's theorem gives the polynomial formula $$V_E(z) = \sup\set{\tfrac{1}{\deg p} \log\abs{p(z)} : p \text{ a polynomial},\ \deg p \ge 1,\ \sup_E \abs{p} \le 1} .$$ Example: for the closed unit ball $E = \set{\abs{z} \le 1}$, $V_E(z) = \max(\log\abs{z}, 0)$. Check: this function lies in $\mathcal{L}$ and vanishes on $E$, so $V_E \ge \max(\log\abs{z}, 0)$. Conversely, let $u \in \mathcal{L}$ with $u \le 0$ on $E$, and let $w$ be a unit vector. The function $\lambda \mapsto u(\lambda w) - \log\abs{\lambda}$ is subharmonic and bounded above on $\abs{\lambda} > 1$, so it extends across $\infty$. It is at most $0$ on $\abs{\lambda} = 1$, so by the maximum principle $u(\lambda w) \le \log\abs{\lambda}$ for $\abs{\lambda} > 1$. Non-example: if $E$ is pluripolar, then $V_E^* \equiv +\infty$, so $V_E^*$ is not in $\mathcal{L}$; for compact $E$ the converse holds as well, and $V_E^* \in \mathcal{L}^+$ when $E$ is compact and not pluripolar. For $n = 1$ and compact $E$ of positive logarithmic capacity, $V_E^*$ is the Green function of the unbounded component of $\C \setminus E$ with pole at infinity, extended by $0$. When $V_E$ is continuous, $E$ is called $L$-regular.