Abstract: 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.

Key words: Brunn-Minkowski inequality, Minkowski addition, Minkowski inequality, Orlicz addition, Quermassintegral