Applications Of Probability Measures To Convex Geometric Analysis

The researches of this thesis belong to the theory of convex bodies in geometric 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 probability measures to convex geometric analysis. These problems have been attracted increased interest for this direction, which refer to the centroid inequality for probability measures, Shephard problem with respect to Gaussian measures, some inequalities for functions and the Erd(?)s-Szekeres theorem with monotone convex sequences.Centroid bodies are special sets which play an important role in the theory of convex geometry, they have been found widely uses in the fields of information theory and analysis. While the centroid inequality is proved to be one of the most useful affine isoperimetric inequalities. In chapter 2, we first give the definitions of the generalized(Orlicz) centroid body for probability measures, and show that the newly defined generalized centroid body is a convex body. Then we establish the centroid inequality by the law of large numbers and the methods of approximation and limiting arguments. Choosing proper values of the density functions and Orlicz functions, we can make the generalized centroid body be the classical centroid body, Lpcentroid body, Orlicz centroid body, mean zonoid and so on.Especially, the concepts of centroid body and mean zonoid are unified through the generalized definition, they are special cases of the generalized centroid body.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(Lp, Orlicz) centroid inequality to compact sets. Moreover,some special asymmetric convex bodies can be obtained.The research on the geometric properties of sections and projections of convex bodies very important significance, and it has been to be one of the hot topics of convex geometry. The related problems are the well known Busemann-Petty problem and Shephard problem. In Chapter 4, we discuss the Shephard problem for Gaussian measures. We prove the answer of the Gaussian analog of the Shephard problem is negative if n ≥ 3 by using a counterexample to the original Shephard problem. The results are consistent with the classical problem.In Chapter 5, we focus on some functions on C+(Sn-1) 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-Santal’o type inequality for functions by studying the relationship between f and f?, the dual function of f.In Chapter 6, we study on the well known Erd(?)s-Szekeres theorem with monotone convex sequences, and we get the minimum value n = n(r, s). For given r, s, we show that any sequence of length n contains a monotone convex subsequence of length r or a monotone concave subsequence of length s.