Applications Of Modern Convex Geometry To Information Theory And Probability Theory

In this paper, we are mainly concerned with the study of applications of modern convex geometry to information theory and probability theory. This article belongs to the Brunn-Minkowski theory, which has been a high-speed developing geometry branch during the past several decades. This thesis works for applied study on convex geometry by using the theory of geometry analysis, ways of integral trans-formations and analysis inequalities.In the second chapter, we establish a strong law of large numbers for the har-monic P-combinations of random star bodies. Starting from this theorem, we prove a strong law of large numbers in Lp space and provide the probabilistic version of dual Brunn-Minkowski inequality.The third chapter establishs a strong law of large numbers on Firey combination in Lp space with two different methods. By applying this theorem, we provide the probabilistic version of quermassintegrals'Brunn-Minkowski inequality on Firey combination.The content of chapter 4 is the general Shapley-Folkman-Starr Theorem. We first introduce a new concept—p-sum of two vectors for nonempty, compact subsets in Rd, which coincides with the usual vector addition in the case p= 1. Then we give some properties of the p-sum. Further we establish the p-type Shapley-Folkman-Starr Theorem.In the fifth chapter, we first introduce a new concept—radial p-th moment of a random vector for star body, which is a general form of the standard p-th moment. Further we establish some properties of the radial p-th moment and give some related applications.In the last chapter, we give A-Renyi entropy power and radial p-th moment of the generalized Gaussian—bG(?) whose contour body is K. Further we obtain some related applications by the relation between radial p-th moment and?-Renyi entropy power.