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

Niufa Fang; Sudan Xing; Deping Ye

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arXiv · 2006.16959

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

Niufa Fang·Sudan Xing·Deping Ye

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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.

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BibTeX RIS

Niufa Fang, Sudan Xing, Deping Ye. 2020-06-30. Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem. https://arxiv.org/abs/2006.16959

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