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  "body": "\n## Statement\n\nLet $\\mu$ and $\\nu$ be Borel probability measures on $\\R^n$ with finite second moments, $\\int \\abs{x}^2 \\, d\\mu \u003c \\infty$ and $\\int \\abs{y}^2 \\, d\\nu \u003c \\infty$.\nSuppose that $\\mu$ is [[absolutely continuous|P0P21KDV6E]] with respect to Lebesgue measure.\nThen:\n\n1. There is a [[convex function|07Q7CNM5WF]] $\\varphi \\colon \\R^n \\to \\R \\cup \\set{+\\infty}$, differentiable at $\\mu$-almost every point, such that the [[pushforward|3N66TM3TSQ]] of $\\mu$ by its [[gradient|0ABK9V9Y64]] is $\\nu$: $(\\nabla \\varphi)_\\# \\mu = \\nu$.\n2. The [[coupling|HTR7J0W66V]] $\\pi = (\\id, \\nabla \\varphi)_\\# \\mu$ is the unique coupling of $\\mu$ and $\\nu$ that minimizes $\\int \\abs{x - y}^2 \\, d\\pi(x, y)$.\n3. The map $\\nabla \\varphi$ is unique: if $\\psi$ is another convex function with $(\\nabla \\psi)_\\# \\mu = \\nu$, then $\\nabla \\psi = \\nabla \\varphi$ $\\mu$-almost everywhere.\n\n## Remarks\n\nYann Brenier proved the theorem in work on the polar factorization of vector fields, published in 1987 and 1991.\nThe proof uses Kantorovich duality: since $\\abs{x - y}^2 = \\abs{x}^2 + \\abs{y}^2 - 2 \\ip{x}{y}$, an optimal plan maximizes $\\int \\ip{x}{y} \\, d\\pi$, and its support lies in the subdifferential of a convex function $\\varphi$.\nA convex function is differentiable outside a Lebesgue null set.\nSo absolute continuity of $\\mu$ makes the subdifferential single-valued $\\mu$-almost everywhere, and the plan is carried by the graph of $\\nabla \\varphi$.\nExample: if $\\nu$ is the image of $\\mu$ under $x \\mapsto A x + b$ with $A$ symmetric positive definite, then $\\varphi(x) = \\frac12 \\ip{Ax}{x} + \\ip{b}{x}$ works, and the affine map is optimal.\nAbsolute continuity cannot be dropped: if $\\mu = \\delta_0$ and $\\nu$ is not a Dirac mass, no map pushes $\\mu$ to $\\nu$.\nWhen $\\mu$ and $\\nu$ have densities $f$ and $g$, the potential $\\varphi$ solves the [[Monge–Ampère equation|14AKAC0S0T]] $g(\\nabla \\varphi) \\det D^2 \\varphi = f$ in a weak sense.\n\n## Explanations\n\n### L1 human\n\nTo move one distribution of mass in space onto another at the least total squared distance, there is a best plan, and it moves each point to a single destination.\nBrenier's theorem says that this best map is the gradient of a convex function, the higher-dimensional version of an increasing function.\nIn one dimension it is the map that sends the left part of one distribution to the left part of the other, in order.\n\n### L2 human\n\nThe theorem solves the [[Monge problem|K8EBASR5VG]] for the quadratic cost.\nIt gives the polar factorization: a nondegenerate map $u$ pushing $\\mu$ to $\\nu$ factors as $u = \\nabla \\varphi \\circ s$, where $\\varphi$ is convex and $s$ preserves $\\mu$.\nIt is the basis of the Wasserstein geometry of probability measures, of displacement convexity, and of regularity theory for the Monge–Ampère equation through Caffarelli's work.\n",
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