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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 $K$, its *logarithmic energy* is\n$$I(\\mu) = \\int_K \\int_K \\log\\frac{1}{\\abs{z - w}}\\, d\\mu(z)\\, d\\mu(w) \\in (-\\infty, \\infty].$$\nThe *Robin constant* of $K$ is $V(K) = \\inf_\\mu I(\\mu)$, the infimum over all Borel probability measures $\\mu$ on $K$.\nThe *logarithmic capacity* of $K$ is $\\operatorname{cap}(K) = e^{-V(K)}$, with $\\operatorname{cap}(K) = 0$ when $V(K) = \\infty$.\n\n## Remarks\n\nExample: for the closed disc or the circle of radius $r$, the minimizing measure is normalized arc length on the circle of radius $r$, and $\\operatorname{cap} = r$.\nFor a segment of length $L$, $\\operatorname{cap} = L/4$.\nA finite set $K$ has $\\operatorname{cap}(K) = 0$: every probability measure on $K$ has an atom, and an atom of mass $m \u003e 0$ at $a$ contributes $m^2 \\log(1/\\abs{a - a}) = \\infty$ to $I(\\mu)$.\nLet $K$ be connected with more than one point, and let its complement in the Riemann sphere $\\hat\\C$ be simply connected.\nLet $\\varphi$ be the conformal map of $\\hat\\C \\setminus K$ onto $\\set{\\abs{z} \u003e 1} \\cup \\set{\\infty}$ with $\\varphi(\\infty) = \\infty$.\nNear $\\infty$ it has the form $\\varphi(z) = z/c + O(1)$ with $c \u003e 0$, and then $\\operatorname{cap}(K) = c$, the outer conformal radius of $K$.\nBy theorems of Fekete and Szegő, the capacity also equals the transfinite diameter of $K$ and its Chebyshev constant.\n",
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