Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem

TL;DR

This paper develops an $L_p$ geometric theory for log-concave functions, extending classical convex geometry concepts to a functional setting, and introduces an $L_p$ Minkowski problem with existence results.

Contribution

It introduces an $L_p$ theory for log-concave functions, including inequalities, surface area measures, and the $L_p$ Minkowski problem, extending convex geometric analysis.

Findings

Established an $L_p$ Minkowski type inequality for log-concave functions.

Defined an $L_p$ surface area measure for log-concave functions.

Proved existence of solutions to the $L_p$ Minkowski problem under mild conditions.

Abstract

The aim of this paper is to develop a basic framework of the $L_p$ theory for the geometry of log-concave functions, which can be viewed as a functional "lifting" of the $L_p$ Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the $L_p$ Asplund sum of log-concave functions for all $p>1$ and the total mass, we obtain a Pr\'ekopa-Leindler type inequality and propose a definition for the first variation of the total mass in the $L_p$ setting. Based on these, we further establish an $L_p$ Minkowski type inequality related to the first variation of the total mass and derive a variational formula which motivates the definition of our $L_p$ surface area measure for log-concave functions. Consequently, the $L_p$ Minkowski problem for log-concave functions, which aims to characterize the $L_p$ surface area measure for log-concave functions, is introduced. The existence of solutions to the $L_p$ Minkowski problem for log-concave functions is obtained for $p>1$ under some mild conditions on the pre-given Borel measures.