= Prékopa–Leindler inequality
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{title2=$\int h\geq(\int f)^{1-\theta}(\int g)^\theta$}
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If $0<\theta<1$ and nonnegative <measurable functions> satisfy $h((1-\theta)x+\theta y)\geq f(x)^{1-\theta}g(y)^\theta$ for all $x,y\in\mathbb R^n$, then their <Lebesgue integrals> obey
$$
\int h\geq\left(\int f\right)^{1-\theta}\left(\int g\right)^\theta.
$$
The <quantile derivative identity> and <monotone substitution inequality> prove the one-dimensional case; <Tonelli theorem> extends it by induction. It implies the <Brunn–Minkowski inequality>.
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