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 $$L_p$$ 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 Prékopa-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.