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  "body": "\n## Definition\n\nLet $E \\subseteq \\C^n$ be a nonempty set and let $\\mathcal{L}$ be the [[Lelong class|X4ZHQG9995]] of $\\C^n$.\nThe *Siciak extremal function* of $E$ is the function $V_E \\colon \\C^n \\to [0, +\\infty]$ given by\n$$V_E(z) = \\sup\\set{u(z) : u \\in \\mathcal{L},\\ u \\le 0 \\text{ on } E} .$$\nIts upper semicontinuous regularisation is $V_E^*(z) = \\limsup_{w \\to z} V_E(w)$.\n\n## Remarks\n\nThe value is at least $0$, since $u = 0$ is admissible.\nFor compact $E$, Siciak's theorem gives the polynomial formula\n$$V_E(z) = \\sup\\set{\\tfrac{1}{\\deg p} \\log\\abs{p(z)} : p \\text{ a polynomial},\\ \\deg p \\ge 1,\\ \\sup_E \\abs{p} \\le 1} .$$\nExample: for the closed unit ball $E = \\set{\\abs{z} \\le 1}$, $V_E(z) = \\max(\\log\\abs{z}, 0)$.\nCheck: this function lies in $\\mathcal{L}$ and vanishes on $E$, so $V_E \\ge \\max(\\log\\abs{z}, 0)$.\nConversely, let $u \\in \\mathcal{L}$ with $u \\le 0$ on $E$, and let $w$ be a unit vector.\nThe function $\\lambda \\mapsto u(\\lambda w) - \\log\\abs{\\lambda}$ is subharmonic and bounded above on $\\abs{\\lambda} \u003e 1$, so it extends across $\\infty$.\nIt is at most $0$ on $\\abs{\\lambda} = 1$, so by the maximum principle $u(\\lambda w) \\le \\log\\abs{\\lambda}$ for $\\abs{\\lambda} \u003e 1$.\nNon-example: if $E$ is [[pluripolar|N0MAGPW8AP]], 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.\nFor $n = 1$ 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 $0$.\nWhen $V_E$ is continuous, $E$ is called $L$-regular.\n",
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