The Correspondence Between The Extremun Problems From Convex Geometry And Information Theory

Convex geometry analysis and Information theory, as two old important subject, developed independent with each other. A vital breakthrough comes from Cover and Thomas's work. They firstly proved the relationship between Brunn-Minkowski inequality and Entropy power inequality, and also provided the public proofs of Brunn-Minkowski inequality and Entropy power inequality. And then, relationship between surface area and Entropy has also been constructed. Those works, which clarified connection between convex geometry analysis and information theory, cause a great deal of research. Given the extensive application of information theory, the importance of this connection is self-evidence. As is well known, isoperimetric inequality plays a vitally important role in analysis and geometry. As a famous example, we can consider the equivalent relation between classical isoperimetric inequality and sobolev inequality. In fact, the equivalence between isoperimetric inequality and Fisher information inequality still holds. This discovery will deeply influence the intrinsic connection between convex geometry analysis and information theory. This is still the research motivation of this paper. We will discuss some intensive inequality in information theory and explore its applications.The rest of this paper is arranged as follows. Chapter 1 briefly introduces the history of convex geometry analysis and information theory, current research situation and connection between those two subject are also presented. Lutwak, Yang and Zhang's works are reviewed in Chapter 2. They proved that the result of contoured distribution is corresponding to the result of ellipsoid and provided their Entropy-Renyi inequality, Fisher inequality and Cramer-Rao inequality. In Chapter 3, Based on providing the important position of Brunn-Minkowski inequality and Entropy Power inequality in convex geometry analysis and information theory, respectively, we specially introduce a public proof of the relationship between Brunn-Minkowski inequality and Entropy Power inequality. In Chapter 4, we study that, when 1≤p < n, L p Petty projection inequality exactly equals integral affine- Fisher information inequality.