Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem
Abstract
The aim of this paper is to develop a basic framework of the theory for the geometry of log-concave functions, which can be viewed as a functional "lifting" of the Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the Asplund sum of log-concave functions for all 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 setting. Based on these, we further establish an Minkowski type inequality related to the first variation of the total mass and derive a variational formula which motivates the definition of our surface area measure for log-concave functions. Consequently, the Minkowski problem for log-concave functions, which aims to characterize the surface area measure for log-concave functions, is introduced. The existence of solutions to the Minkowski problem for log-concave functions is obtained for under some mild conditions on the pre-given Borel measures.
Keywords
Cite
@article{arxiv.2006.16959,
title = {Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem},
author = {Niufa Fang and Sudan Xing and Deping Ye},
journal= {arXiv preprint arXiv:2006.16959},
year = {2020}
}