一些几何不等式的等价关系

Equivalence properties of some geometric inequalities

Brunn-Minkowski不等式和Minkowski不等式是凸几何中的两个重要而基本的不等式.近期,已有学者得到了这两个不等式的Orlicz版本,从而构建起Orlicz-Brunn-Minkowski理论的框架.本工作证明经典的Brunn-Minkowski不等式、Minkowski不等式、Orlicz-BrunnMinkowski不等式和Orlicz-Minkowski不等式是等价的.

Brunn-Minkowski inequality and Minkowski inequality are two important and fundamental inequalities in convex geometric analysis. Recently, some researchers established Orlicz extension of these two inequalities, and constructed a general framework for the Orlicz-Brunn-Minkowski theory. The purpose of this paper is to show equivalence properties of these four inequalities, i.e., classical Brunn-Minkowski inequality, classical Minkowski inequality, Orlicz-Brunn-Minkowski inequality and Orlicz-Minkowski inequality.