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  "body": "\n## Statement\n\nLet $n \\ge 1$ and $0 \u003c t \u003c 1$.\nLet $f, g, h \\colon \\R^n \\to [0, \\infty)$ be Lebesgue [[measurable functions|YYNCJSR0PJ]] such that\n$$h((1-t) x + t y) \\ge f(x)^{1-t} g(y)^{t} \\quad \\text{for all } x, y \\in \\R^n.$$\nThen\n$$\\int_{\\R^n} h \\, dx \\ge \\left(\\int_{\\R^n} f \\, dx\\right)^{1-t} \\left(\\int_{\\R^n} g \\, dx\\right)^{t}.$$\n\n## Remarks\n\nThe integrals are [[Lebesgue integrals|DWJF7ABPR1]] and may be infinite.\nThe proof for $n = 1$ uses a transport map between $f$ and $g$ after normalisation, and the general case follows by induction on $n$ with Fubini's theorem.\nExample: let $A, B$ be nonempty compact sets and $f = 1_A$, $g = 1_B$, $h = 1_{(1-t)A + tB}$.\nThe hypothesis holds, and the conclusion gives $\\lambda((1-t) A + t B) \\ge \\lambda(A)^{1-t} \\lambda(B)^{t}$.\nThis multiplicative form implies the [[Brunn–Minkowski theorem|9B02RPN7XD]] by scaling $A$ and $B$.\nIt is a reversed form of Hölder's inequality: Hölder bounds $\\int f^{1-t} g^t$ from above, while here a function above the \"sup-convolution\" is bounded from below.\n\n## Explanations\n\n### L1 human\n\nIf a function $h$ is large at each weighted midpoint of points where $f$ and $g$ are large, then the integral of $h$ is large too.\nIt is the functional version of the Brunn–Minkowski inequality for volumes of sums of sets.\n\n### L2 human\n\nA main consequence is that log-concave functions and measures are stable: marginals and convolutions of log-concave densities are log-concave.\nThis makes the inequality a basic tool in convex geometry and in the concentration of measure, for example for Gaussian measures.\n",
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