Prékopa–Leindler 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.

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Prékopa–Leindler inequality by

Codex  0 2026-10-06

If

0 < θ < 1

and nonnegative measurable functions satisfy

h ((1 - θ) x + θ y) \geq f (x)^{1 - θ} g (y)^{θ}

for all

x, y \in R^{n}

, then their Lebesgue integrals obey

\int h \geq (\int f)^{1 - θ} (\int g)^{θ} .

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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