The Extreme Problem For Projection Of Convex Bodies And Its Related Inequality

The researches of this thesis belong to the theory of convex bodies in geomet-ric analysis,which is also called convex geometry or convex geometric analysis for short.The Brunn-Minkowski theory,also called mixed volume theory,is the core part of this theory.This dissertation is devoted to the researches of the applications of the problem of projection on convex bodies to convex geometric analysis.These problems have been attracted increased interest for this direction,which refer to the inequality for dual Minkowski type inequality,the asymmetric version of the generalized?Orlicz?centroid body for probability measures,some convex inequalities for functions and the Orlicz-Brunn-Minkowski inequalities for polar bodies and dual star bodies.Projection on convex bodies is always one of hot spot to convex geometric analysis.In Chapter 2 of this paper,we study the dual problems of the Orlicz Brunn-Minkowski theory.We give the definition of the dual Orlicz dual mixed volume.It is the generalization of dual L_p mixed volume.We also study the properties of the dual Orlicz mixed volume.In Chapter 3,we consider the asymmetric version of the generalized?Orlicz?centroid body for probability measures,and the asymmetric centroid inequality for the generalized centroid body is also obtained.The idea is up to the methods of probability and limiting arguments used by Paouris and Pivovarov.Choosing proper values of the density functions and Orlicz functions,one can generalize the asymmetric classical?L_p,Orlicz?centroid inequality to compact sets.Moreover,some special asymmetric convex bodies can be obtained.In Chapter 4,we focus on some functions on C⁺(S^n-1)andGL?n?to obtain some properties and inequalities.First we define the volume and surface area of the function f by the Aleksandrov body with respective to f,and we get a surface area formula of the function f.Then we establish the Blaschke-Santaló type inequality for functions by studying the relationship between f and f^°,the dual function of f,get the affine isoperimetric inequality and Blaschke-Santaló type inequality.In chapter 5,we establish the Orlicz-Brunn-Minkowski inequalities for polar bodies and dual star bodies.These results can be considered as“polar”counterparts of the existing Orlicz-Brunn-Minkowski inequality and its dual.