The Orlicz Theory And Measures Of Asymmetry For Convex Bodies

The content of this thesis concerns mainly with the Orlicz-Brunn-Minkowski theory, which is a high-speed developing geometry branch after2010. The thesis is devoted to some basic concepts of the Orlicz-Brunn-Minkowski theory, such as Orlicz metric, Orlicz dual mixed volume, Orlicz difference body and Orlicz measure of asymmetry etc. In addition, some basic properties, such as the asymmetry, of convex bodies of constant width are studied as well.This thesis consists of two parts. The first part contains Charpter â…¡, â…¢, â…£ and â…¤, where we focus on some basic topics on Orlicz-Brunn-Minkowski theory. The second part contains Chapter VI and VII, where we study the asymmetry of convex bodies of constant width.Concretely, in Charpter II, we study the Orlicz metric, which is an extension of classic Hausdorff metric and we prove that the space Cn of compact convex subsets of Rn equipped with Orlicz metrics is complete, separable. In Charpter III, in terms of the notion of Orlicz addition, we introduce the Orlicz difference body, and obtain an Orlicz difference body inequality in R2by the method of shadow system. In charpter IV, we introduce the concepts of harmonic Orlicz combination and Orlicz dual mixed volume, and get the Orlicz-Minkowski dual mixed volume inequality and the dual Orlicz-Brunn-Minkowski inequality. In addition, we also introduce the Orlicz dual mixed quermassintegrals, and get the corresponding inequalities. In charpter V, using the concept of Orlicz mixed volume, we introduce the notion of Orlicz measure of asymmetry, which is an extension of the classic Minkowski measure of asymmetry.In the second part, Charpter VI devotes to the Minkowski measure of asymmetry for convex bodies of constant width, where we show that the completions of a regular simplex are the most asymmetric among all convex bodies of constant width. In addition, we get also a stability theorem on the Minkowski measure of asymmetry for2-dimensional convex bodies of constant width. Charpter VII concerns with the the mean Minkowski measure of asymmetry of the convex bodies of constant width, where we discover that the completions of a regular simplex are the most asymmetric as well among all convex bodies of constant width with respect to the mean Minkowski measure of asymmetry, which gives some hints for the open problem on the minimal volume of convex bodies of constant width.