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  "body": "\n## Statement\n\nLet $K$ be a [[compact set|0R6EWFKRCF]] in $\\C$ with positive logarithmic capacity.\nLet $\\nu$ be its [[equilibrium measure|464NN2K4FZ]] and let $V(K)$ be its [[Robin constant|W3NVXBEMRQ]].\nLet $U^\\nu(z) = \\int \\log \\frac{1}{\\abs{z - w}} \\, d\\nu(w)$ be the [[logarithmic potential|46NHWJHW44]] of $\\nu$.\nThen:\n\n1. $U^\\nu(z) \\le V(K)$ for every $z \\in \\C$.\n2. $U^\\nu(z) = V(K)$ for every $z \\in K \\setminus E$, where $E$ is an $F_\\sigma$ subset of the boundary $\\partial K$ of [[capacity zero|W9N1TR1X8P]].\n\n## Remarks\n\nProof idea: suppose that $U^\\nu \u003c V(K)$ on a compact subset of $K$ of positive capacity, or $U^\\nu \u003e V(K)$ on a set of positive $\\nu$-measure.\nThen moving a little mass of $\\nu$ lowers the energy, which contradicts minimality.\nThis gives $U^\\nu \\ge V(K)$ on $K$ outside a set of capacity zero and $U^\\nu \\le V(K)$ on the support of $\\nu$.\nThe maximum principle for potentials then extends the upper bound from the support of $\\nu$ to all of $\\C$.\nA separate argument shows that the support of $\\nu$ lies in $\\partial K$.\n\nExample: for the unit circle, $\\nu = d\\theta / 2\\pi$ and $U^\\nu(z) = -\\log \\max(\\abs{z}, 1)$.\nThis is $0 = V(K)$ on the closed unit disc and negative outside, as the theorem says.\nThe exceptional set $E$ cannot be dropped in general: it can contain boundary points that are irregular for the Dirichlet problem, where the potential is strictly below $V(K)$.\n\n## Explanations\n\n### L1 human\n\nWhen a unit charge sits in equilibrium on a conductor $K$, the voltage it creates is the same at almost every point of $K$.\nNowhere in the plane is the voltage higher than this value.\nThe only exceptions on $K$ form a set so thin that it has capacity zero.\n\n### L2 human\n\nFrostman's theorem is the bridge between the energy definition of capacity and the Green function.\nOn the unbounded component $\\Omega$ of $\\C \\setminus K$, the function $g(z) = V(K) - U^\\nu(z)$ is the Green function of $\\Omega$ with pole at infinity.\nThe exceptional set $E$ is linked to the boundary points that are irregular for the Dirichlet problem.\nThe Kellogg–Evans theorem says that such points form a set of capacity zero.\nThe theorem is also the basic tool for computing capacities: a probability measure on $K$ whose potential is constant on all of $K$ must be the equilibrium measure.\n",
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