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  "body": "\n## Definition\n\nLet $K$ be a [[compact set|0R6EWFKRCF]] in $\\C$.\nFor a Borel [[probability measure|059TJN0NZ1]] $\\mu$ on $K$, let\n$$I(\\mu) = \\int_K \\int_K \\log \\frac{1}{\\abs{z - w}} \\, d\\mu(z) \\, d\\mu(w)$$\nbe its logarithmic [[energy|SFSRQCM3EG]], with values in $(-\\infty, \\infty]$.\nThe *Robin constant* of $K$ is\n$$V(K) = \\inf \\set{ I(\\mu) : \\mu \\text{ a Borel probability measure on } K },$$\nwith $V(K) = \\infty$ when $I(\\mu) = \\infty$ for every such $\\mu$.\n\n## Remarks\n\nThe logarithmic capacity of $K$ is $\\operatorname{cap}(K) = e^{-V(K)}$.\nIf $V(K) \u003c \\infty$ and $\\Omega$ is the unbounded component of $\\hat\\C \\setminus K$, the Green function of $\\Omega$ with pole at infinity satisfies $g_\\Omega(z, \\infty) = \\log \\abs{z} + V(K) + o(1)$ as $z \\to \\infty$.\n\nExample: let $K$ be the circle of radius $r$ centred at $0$ and let $\\mu$ be normalized arc length on it.\nFor $\\abs{z} = r$ the mean value of $\\log(1/\\abs{z - w})$ over the circle is $-\\log r$, so $I(\\mu) = -\\log r$.\nThis $\\mu$ is the minimizer, so $V(K) = -\\log r$.\nThe Green function $\\log(\\abs{z}/r)$ of the outside of the circle agrees with the expansion above.\n\nNon-example of a finite value: for $K = \\set{0}$ the only probability measure is $\\delta_0$, and $I(\\delta_0) = \\log(1/0) = \\infty$, so $V(K) = \\infty$.\n\nSome authors use the opposite sign or the energy kernel $\\log \\abs{z - w}$, so that the Robin constant becomes $\\log \\operatorname{cap}(K)$.\n",
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