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  "body": "\n## Definition\n\nLet $\\mathcal{P}_2^{\\mathrm{ac}}(\\R^n)$ be the set of Borel probability measures on $\\R^n$ with finite second moment that are [[absolutely continuous|P0P21KDV6E]] with respect to Lebesgue measure.\nFor $\\mu_0, \\mu_1 \\in \\mathcal{P}_2^{\\mathrm{ac}}(\\R^n)$, let $\\nabla \\varphi$ be the optimal map from $\\mu_0$ to $\\mu_1$ given by [[Brenier's theorem|CVQVBMHH4X]].\nThe *displacement interpolation* between them is the curve\n$$\\mu_t = \\left( (1 - t) \\id + t \\nabla \\varphi \\right)_\\# \\mu_0, \\qquad t \\in [0, 1],$$\nwhere $_\\#$ denotes the [[pushforward|3N66TM3TSQ]].\nA functional $F \\colon \\mathcal{P}_2^{\\mathrm{ac}}(\\R^n) \\to (-\\infty, +\\infty]$ is *displacement convex* if, for all $\\mu_0, \\mu_1 \\in \\mathcal{P}_2^{\\mathrm{ac}}(\\R^n)$, the function $t \\mapsto F(\\mu_t)$ is convex on $[0, 1]$.\n\n## Remarks\n\nExample: let $V \\colon \\R^n \\to \\R$ be [[convex|07Q7CNM5WF]] and $F(\\mu) = \\int V \\, d\\mu$.\nThen $F(\\mu_t) = \\int V\\big((1 - t) x + t \\nabla \\varphi(x)\\big) \\, d\\mu_0(x)$, and the integrand is convex in $t$ for each $x$, so $F$ is displacement convex.\n\nNon-example: on $\\R$, let $V(x) = -x^2$, let $\\mu_0$ be uniform on $[0, 1]$ and $\\mu_1$ uniform on $[2, 3]$.\nThe optimal map is $x \\mapsto x + 2$, so $\\mu_t$ is uniform on $[2t, 2t + 1]$ and $F(\\mu_t) = -\\int_0^1 (x + 2t)^2 \\, dx = -\\left(\\tfrac13 + 2t + 4t^2\\right)$, which is strictly concave in $t$.\n\nThe interpolation $\\mu_t$ is a constant-speed geodesic for the Wasserstein distance $W_2$, so displacement convexity is convexity along $W_2$-geodesics.\nThe functional $F$ is *$\\lambda$-displacement convex*, for a real $\\lambda$, if $t \\mapsto F(\\mu_t) - \\frac{\\lambda}{2} t^2 W_2(\\mu_0, \\mu_1)^2$ is convex.\nMcCann showed that the entropy $\\int \\rho \\log \\rho \\, dx$ of a density $\\rho$ is displacement convex on $\\R^n$.\nThe interaction energy $\\iint W(x - y) \\, d\\mu(x) \\, d\\mu(y)$ with $W$ convex is also displacement convex.\nIt need not be convex in the usual sense: for $W(z) = \\abs{z}^2$ it equals twice the variance of $\\mu$, which is concave under mixtures $(1 - t)\\mu_0 + t\\mu_1$.\n",
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