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  "body": "\n## Summary\n\nThe Riesz $s$-energy of a positive measure $\\mu$ on $\\R^d$ is the double integral of the kernel $\\abs{x - y}^{-s}$ against $\\mu \\times \\mu$.\nFor $s = d - 2$ with $d \\ge 3$ it is the Newtonian [[energy of a measure|SFSRQCM3EG]], and for other $s$ it is the energy attached to the [[Riesz potential|1V2N8MFSHC]].\nMinimising this energy over probability measures on a compact set defines the Riesz capacity of the set.\nThe discrete version, a sum over pairs of points, is the energy in problems on optimal point configurations such as the [[Thomson problem|3C7V5X5FY8]].\n\n## Explanations\n\n### L1 human\n\nSpread a unit of charge over a set, and let two small pieces of charge at distance $r$ repel each other with strength $r^{-s}$.\nThe Riesz $s$-energy adds up this repulsion over all pairs of pieces.\nCharge piled up at a single point gives infinite energy, so a set can carry a measure of finite energy only if it is large enough.\nFor example, the uniform probability measure on the unit sphere of $\\R^3$ has $1$-energy equal to $1$.\nThe exponent $s$ tunes how strongly close pairs are penalised.\n",
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