论文标题
log-concave函数的几何形状：$ l_p $ asplund sum和$ l_ {p} $ minkowski问题
Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem
论文作者
论文摘要
本文的目的是为log-conconcave函数的几何形状开发一个基本框架，可以将其视为$ l_p $ brunn-minkowski理论的功能“提升”。为了实现这一目标，通过将$ l_p $ asplund的log-conconconcave函数和所有$ p> 1 $和总质量结合在一起，我们获得了prékopa-ledindler型不平等，并提出了$ L_P $设置中总质量的第一个变化的定义。基于这些，我们进一步建立了与总质量的第一个变化相关的$ L_P $ Minkowski类型不等式，并得出了一个变异公式，该公式激发了我们$ L_P $表面积的定义。因此，引入了$ l_p $ minkowski的log-concove函数问题，旨在表征$ l_p $ conconconconcove函数的$ L_P $表面积测量。在某些温和的条件下，在预先授予的博雷尔措施的某些温和条件下，获得了$ L_P $ Minkowski问题的解决方案。
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.
