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  "body": "\n## Statement\n\nLet $(M, d)$ be a [[Polish space|584GXXY73M]].\nLet $\\mu$ and $\\nu$ be Borel [[probability measures|059TJN0NZ1]] on $M$ with finite first moments, that is, $\\int_M d(x, x_0) \\, d\\mu(x) \u003c \\infty$ and $\\int_M d(x, x_0) \\, d\\nu(x) \u003c \\infty$ for some $x_0 \\in M$.\nLet $W_1$ be the $1$-[[Wasserstein distance|VEY29CS26X]].\nThen\n$$W_1(\\mu, \\nu) = \\sup \\set{ \\int_M f \\, d\\mu - \\int_M f \\, d\\nu },$$\nwhere the [[supremum|0R3H5PANRQ]] is over all functions $f \\colon M \\to \\R$ with $\\abs{f(x) - f(y)} \\le d(x, y)$ for all $x, y \\in M$.\n\n## Remarks\n\nThe inequality $\\ge$ is easy.\nFor a coupling $\\gamma$ and a $1$-[[Lipschitz|0AW1E6YTJY]] $f$, one has $\\int f \\, d\\mu - \\int f \\, d\\nu = \\int (f(x) - f(y)) \\, d\\gamma(x, y) \\le \\int d(x, y) \\, d\\gamma(x, y)$.\nThe reverse inequality is a case of Kantorovich duality in optimal transport, for the cost $c(x, y) = d(x, y)$.\nFor this cost the dual potentials can be taken $1$-Lipschitz, with the second potential equal to $-f$.\n\nExample: for Dirac measures $\\delta_x$ and $\\delta_y$, the function $f = d(\\cdot, y)$ is $1$-Lipschitz and gives $f(x) - f(y) = d(x, y)$.\nThis matches $W_1(\\delta_x, \\delta_y) = d(x, y)$.\n\nThe moment condition makes all the integrals finite, since a $1$-Lipschitz $f$ satisfies $\\abs{f(x)} \\le \\abs{f(x_0)} + d(x, x_0)$.\n\n## Explanations\n\n### L1 human\n\nThe cheapest cost of moving one mass distribution onto another can be computed in a second way.\nOne looks for a price function that changes by at most the distance between two places.\nThe largest difference of average prices under the two distributions equals the transport cost.\n\n### L2 human\n\nThe duality turns a minimum over couplings into a maximum over Lipschitz functions.\nIt shows that $W_1(\\mu, \\nu)$ depends only on the signed measure $\\mu - \\nu$, as a dual norm for the Lipschitz seminorm.\nThis is the form of $W_1$ used in Wasserstein generative adversarial networks, where the Lipschitz function is a trained critic.\nIt is often called the Kantorovich–Rubinstein theorem.\n",
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