A Pr\'ekopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality

Abstract

In this paper, we prove a Pr\'ekopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Pr\'ekopa-Leindler inequality. In addition, we prove a functional $L_p$ Minkowski inequality.