  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.
