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

TL;DR

This paper extends classical inequalities in convex geometry by establishing a Prékopa-Leindler type inequality for the $L_p$ Brunn-Minkowski inequality, and introduces a functional $L_p$ Minkowski inequality.

Contribution

It introduces a new inequality that generalizes the Prékopa-Leindler inequality within the $L_p$ Brunn-Minkowski framework, expanding the theoretical understanding of convex geometric inequalities.

Findings

Proves a Prékopa-Leindler type inequality for $L_p$ Brunn-Minkowski.

Recovers classical Prékopa-Leindler inequality as a special case.

Establishes a functional $L_p$ 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.