Siciak extremal function

Summary

The Siciak extremal function $V_E$ of $E \subseteq \C^n$ is the supremum of all functions of the Lelong class that are at most $0$ on $E$. For compact $E$ it can also be computed from polynomials. It is the supremum of $\frac{1}{\deg p} \log\abs{p}$ over polynomials $p$ of degree at least $1$ with $\abs{p} \le 1$ on $E$. It is the higher-dimensional version of the Green function of a planar compact set with pole at infinity. It is also called the pluricomplex Green function of $E$ with pole at infinity. It controls polynomial approximation and interpolation on $E$, and for compact $E$ its upper semicontinuous regularisation is identically $+\infty$ exactly when $E$ is pluripolar.