 Prékopa–Leindler inequality

If $0<\theta <1$ and nonnegative measurable functions 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.

The Prékopa–Leindler inequality is a fundamental result in the field of convex analysis and probability theory. It provides a way to compare the integrals of certain convex functions over different sets.