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- W3038823773 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." @default.
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- W3038823773 date "2020-06-30" @default.
- W3038823773 modified "2023-10-16" @default.
- W3038823773 title "Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem" @default.
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- W3038823773 doi "https://doi.org/10.48550/arxiv.2006.16959" @default.
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