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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[Polish spaces|584GXXY73M]], let $\\mu$ be a Borel [[probability measure|059TJN0NZ1]] on $X$ and $\\nu$ a Borel probability measure on $Y$.\nLet $c \\colon X \\times Y \\to [0, \\infty]$ be a lower semicontinuous cost function.\nLet $\\Pi(\\mu, \\nu)$ be the set of [[couplings|HTR7J0W66V]] of $\\mu$ and $\\nu$, the Borel probability measures on $X \\times Y$ with marginals $\\mu$ and $\\nu$.\nThen\n$$\\min_{\\pi \\in \\Pi(\\mu, \\nu)} \\int_{X \\times Y} c \\, d\\pi = \\sup \\set{ \\int_X \\varphi \\, d\\mu + \\int_Y \\psi \\, d\\nu : \\varphi \\in L^1(\\mu), \\ \\psi \\in L^1(\\nu), \\ \\varphi(x) + \\psi(y) \\le c(x, y) \\text{ for all } x, y },$$\nand the minimum on the left is attained.\nThe supremum does not change if $\\varphi$ and $\\psi$ are required to be bounded and continuous.\n\n## Remarks\n\nThe left side is the Kantorovich form of the [[optimal transport problem|K8EBASR5VG]], and the right side is its dual.\nThe inequality $\\ge$ is easy: if $\\varphi(x) + \\psi(y) \\le c(x, y)$ and $\\pi \\in \\Pi(\\mu, \\nu)$, integrating against $\\pi$ gives $\\int \\varphi \\, d\\mu + \\int \\psi \\, d\\nu \\le \\int c \\, d\\pi$.\nThe reverse inequality is a minimax argument.\nFor compact spaces and continuous $c$ it follows from a duality theorem of convex analysis, such as the Fenchel–Rockafellar theorem.\nThe general case follows by approximation.\nExample: if $\\mu = \\delta_a$ is a Dirac mass, the only coupling is $\\delta_a \\otimes \\nu$, so the left side equals $\\int c(a, y) \\, d\\nu(y)$.\nWhen $X = Y$ and $c$ is a metric $d$, the dual can be taken over $\\psi = -\\varphi$ with $\\varphi$ $1$-Lipschitz, which is the [[Kantorovich–Rubinstein duality|SYXSZD8EFK]].\nLower semicontinuity of $c$ is used to show that the minimum is attained, through the weak compactness of $\\Pi(\\mu, \\nu)$.\n\n## Explanations\n\n### L1 human\n\nThink of a shipping company that moves goods from factories with distribution $\\mu$ to shops with distribution $\\nu$, at cost $c(x, y)$ per unit.\nInstead of paying for each route, one may sell the job to a contractor who charges a loading price $\\varphi(x)$ and an unloading price $\\psi(y)$.\nThe contractor's prices are competitive if $\\varphi(x) + \\psi(y)$ never exceeds the direct cost $c(x, y)$.\nThe theorem says that the best competitive income of the contractor equals the least shipping cost.\n\n### L2 human\n\nKantorovich duality is the linear-programming duality of optimal transport, extended to infinite spaces.\nUnder mild integrability conditions on $c$, optimal dual pairs $(\\varphi, \\psi)$ exist and can be chosen $c$-concave, and an optimal plan is concentrated on the set where $\\varphi(x) + \\psi(y) = c(x, y)$.\nThis is the starting point for Brenier's theorem, where the optimal $\\varphi$ gives the optimal map, and for the theory of Wasserstein distances.\n",
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