On Extensions of the Brunn-Minkowski and Prékopa-Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation

Abstract

We extend the Prekopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prekopa—Leindler and Brunn-Minkowski theorems by introducing the notion of essential addition. Our proof of the Prekopa—Leindler theorem is simpler than the original one. We sharpen the inequality that the marginal of a log concave function is log concave, and we prove various moment inequalities for such functions. Finally, we use these results to derive inequalities for the fundamental solution of the diffusion equation with a convex potential.