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.

On the framework of $L_{p}$ summations for functions

“…(Please see definitions for W s p,j , u H and ψ H in Subsection 5.3 for detailed information.) When j = 0 and s = 0, it recovers the integral interpretation of variation formulas in [28] and [54], for p ≥ 1 and 0 < p < 1, respectively.…”

“…We verify that these L p,s convolutions satisfy elegant properties by L p coefficients (C p,λ,t , D p,λ,t ). In Subsection 2.2, we introduce the L p,s Asplund summation through the base functions for s-concave functions inspired by the case of log-concave functions [28,53,54] for p ≥ 1 and 0 < p < 1, respectively. Furthermore in Subsection 2.3, we compare definitions proposed in Subsections 2.1 and 2.2, and prove that for log-concave functions (s = 0), these two summations for p ≥ 1 are equivalent to each other.…”

“…Another type of summations-Asplund summation (or L 1 Asplund summation) for functions using infimal convolution ( ) for base functions, and its L p extensions for log-concave functions [28,53,54] with L p averages for base functions is defined as follows. In the following, we list some basics of Asplund summation for functions first.…”

“…In [28] and [54], the authors proposed the L p summations in terms of the base functions for s = 0 (log-concave functions), and the solutions to the corresponding Minkowski type problems for p ≥ 1 and 0 < p < 1 are also proposed and solved, respectively. Based on these two types of summations for functions above, i.e., the L p supremal-convolution for p ≥ 1, s ≥ 0 and Asplund summation for s-concave functions including log-concave case, we consider more complicated cases of summations for s-concave functions for various cases for p and s.…”

“…Elementary properties are also provided by a detailed analysis for this new sum in Proposition 2.9. Moreover, following the method of L p Asplund summation for log-concave functions when p ≥ 1 [28] and 0 < p < 1 [54], we introduce the L p,s Asplund summation for s-concave functions using L p addition for base functions. Let p > 0.…”