A stochastic Prekopa-Leindler inequality for log-concave functions

TL;DR

This paper establishes a stochastic dominance principle for log-concave functions that underpins the Prékopa-Leindler inequality, extending the probabilistic approach used for convex sets to a broader class of functions.

Contribution

It introduces a stochastic dominance framework for log-concave functions, providing a new probabilistic proof of the Prékopa-Leindler inequality.

Findings

Stochastic dominance explains the Prékopa-Leindler inequality for log-concave functions.

Extends probabilistic methods from convex sets to log-concave functions.

Provides new insights into the structure of inequalities in convex analysis.

Abstract

The Brunn-Minkowski and Pr\'{e}kopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of $\log$-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Pr\'{e}kopa-Leindler inequality.