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  "body": "\n## Definition\n\nLet $d \\ge 1$ be an integer and let $s \u003e 0$.\nThe *Riesz $s$-kernel* on $\\R^d$ is $k_s(x, y) = \\abs{x - y}^{-s}$ for $x \\ne y$, and $k_s(x, x) = +\\infty$, where $\\abs{x - y}$ is the [[Euclidean distance|0NPPGHQNXQ]].\nLet $\\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] on $\\R^d$ with compact support.\nThe *Riesz $s$-energy* of $\\mu$ is\n$$I_s(\\mu) = \\int \\int \\frac{d\\mu(x)\\, d\\mu(y)}{\\abs{x - y}^{s}} \\in [0, +\\infty].$$\nThe measure $\\mu$ has *finite $s$-energy* if $I_s(\\mu) \u003c \\infty$.\nFor distinct points $x_1, \\dots, x_N \\in \\R^d$, the *discrete Riesz $s$-energy* is\n$$E_s(x_1, \\dots, x_N) = \\sum_{i \\ne j} \\frac{1}{\\abs{x_i - x_j}^{s}}.$$\n\n## Remarks\n\nThe inner integral $\\int \\abs{x - y}^{-s}\\, d\\mu(y)$ is, up to a constant factor when $0 \u003c s \u003c d$, the [[Riesz potential|1V2N8MFSHC]] of $\\mu$ at $x$.\nSo $I_s(\\mu)$ is the integral of that potential against $\\mu$.\nExample: let $d = 3$, $s = 1$ and let $\\sigma$ be the uniform probability measure on the unit sphere.\nBy the shell theorem, $\\int \\abs{x - y}^{-1}\\, d\\sigma(y) = 1$ for every $x$ on the sphere, so $I_1(\\sigma) = 1$.\nNon-example of finite energy: a point mass $\\delta_a$ has $I_s(\\delta_a) = k_s(a, a) = +\\infty$ for every $s \u003e 0$.\nFor $s \\ge d$, every nonzero measure as above has infinite $s$-energy.\nThe Riesz $s$-capacity of a compact set $K$ is $1 / \\inf I_s(\\mu)$, the infimum over Borel probability measures $\\mu$ on $K$.\n",
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