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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\mu$ and $\\nu$ be finite positive [[Borel measures|DYR1QFNGNJ]] on $\\R^n$ with compact supports.\nUse the kernel $\\abs{x}^{2-n}$ when $n \\ge 3$ and $\\log(1/\\abs{x})$ when $n = 2$, and assume that the [[energies|SFSRQCM3EG]] $I(\\mu)$ and $I(\\nu)$ are finite.\nWhen $n = 2$, assume also that $\\mu(\\R^2) = \\nu(\\R^2)$.\nThen the [[mutual energy|6VR8GASZJY]] $I(\\mu, \\nu)$ is finite and\n$$I(\\mu - \\nu) = I(\\mu) - 2 I(\\mu, \\nu) + I(\\nu) \\ge 0,$$\nwith equality if and only if $\\mu = \\nu$.\n\n## Remarks\n\nProof idea for $n \\ge 3$: the [[Fourier transform|DPVM34ADZV]] of $\\abs{x}^{2-n}$, as a tempered distribution, is $c_n \\abs{\\xi}^{-2}$ with a constant $c_n \u003e 0$.\nFor $\\lambda = \\mu - \\nu$ this gives\n$$I(\\lambda) = c\\, \\int_{\\R^n} \\frac{\\abs{\\hat{\\lambda}(\\xi)}^2}{\\abs{\\xi}^2}\\, d\\xi$$\nwith a constant $c \u003e 0$, which is nonnegative and vanishes only when $\\hat\\lambda = 0$, that is, $\\mu = \\nu$.\nIn the plane the logarithmic kernel has a similar representation on measures of total mass $0$, which is why the masses must agree.\n\nThe mass condition is necessary in the plane.\nLet $\\mu$ be the uniform probability measure on the circle $\\abs{z} = R$ with $R \u003e 1$, and $\\nu = 0$.\nThen $I(\\mu - \\nu) = I(\\mu) = \\log(1/R) \u003c 0$.\n\nFor $n \\ge 3$ the theorem implies the [[Cauchy–Schwarz inequality|0RFWW56Y22]] $I(\\mu, \\nu)^2 \\le I(\\mu) I(\\nu)$, since $I(\\mu - t\\nu) = I(\\mu) - 2t\\, I(\\mu, \\nu) + t^2 I(\\nu) \\ge 0$ for every $t \u003e 0$ and $I(\\mu, \\nu) \\ge 0$; take $t = I(\\mu, \\nu)/I(\\nu)$ when $I(\\nu) \u003e 0$.\n\n## Explanations\n\n### L1 human\n\nThe energy of a charge distribution with both positive and negative charges is never negative, as long as the total charge is balanced in the plane.\nIt is zero only when there is no charge at all.\nSo energy behaves like the square of a length on the space of charge distributions.\n\n### L2 human\n\nThis is the key fact of the energy method in potential theory.\nIt makes $\\sqrt{I(\\mu - \\nu)}$ a distance on measures of finite energy, and a theorem of Cartan says that positive measures of finite energy are complete for this distance when $n \\ge 3$.\nUniqueness of the equilibrium measure follows: if $\\mu$ and $\\nu$ both minimise the energy among probability measures on a compact set, then the energy of $\\frac12(\\mu + \\nu)$ forces $I(\\mu - \\nu) = 0$, so $\\mu = \\nu$.\n",
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  "name": "Positive definiteness of the energy",
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