A stochastic Prékopa–Leindler inequality for log-concave functions

Abstract

The Brunn–Minkowski and Prékopa–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 [Formula: see text]-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa–Leindler inequality.