Prékopa–Leindler inequality

In the finite, nonzero normalization, let $0<\theta <1$, let $f,g$ be nonnegative integrable functions on ${\mathbb{R}}^n$ with $0<\int f,\int g<\infty$, and let $h$ be a nonnegative measurable function. If they satisfy $h((1-\theta )x+\theta y)\ge f(x{)}^{1-\theta }g(y{)}^{\theta }$ for all $x,y\in {\mathbb{R}}^n$, then their Lebesgue integrals obeyThe quantile derivative identity and monotone substitution inequality prove the one-dimensional case; Tonelli theorem extends it by induction. It implies the Brunn–Minkowski inequality.