On Several Extremal Problems Of Orlicz Brunn-Minkowski Theory

The researches of this thesis belongs to the theory of convex geometric anal-ysis, and devotes to the study of the Orlicz Brunn-Minkowski theory. Our mainworks are to research the theories of convex bodies, some inequalities and extremalproperties of geometry bodies by applying the basic notions, basic theories of OrliczBrunn-Minkowski theory.The purpose of chapter2is to study the Mahler conjecture combining with theOrlicz Brunn Minkowski theory. We first give the definition of the Orlicz zonotopeswhich is the generalization of L_pzonotopes. The notion of the shadow systemswhich first gave by Rogers and Shephard plays an important roles in proving ourmain theorems. The shadow systems of convex bodies can be seen as the shadowsystems of points. We prove the following lemma: Let φ∈C, Λ_t, t∈[t₁, t₂] is ashadow system along the direction of v, then Z_φ（Λ_t） is a shadow system along thesame direction.In chapter3, we study the extremal problems of a new Sylvester type func-tional. Extremal problems are important problems in convex geometry analysisand there are many results about extremal problems. With the development ofthe Orlicz Brunn-Minkowski theory, we give the definition of Orlicz Sylvester typefunctional A（K）. Using the notion of shadow systems, we study the extremal prob-lems of A（K）. We obtain that when K are ellipsoids, A（K） is minimum. In hecase of n=2, A（K） attains its maximum when K is a triangle.The purpose of chapter4is to study the dual problems of the Orlicz Brunn-Minkowski theory. We give the definition of the dual Orlicz dual mixed volume. Itis the generalization of dual L_pmixed volume. We also study the properties of thedual Orlicz mixed volume.The classical John theorem says how far a convex body is from being an ellipsoid. In chapter5, we give the definition of double John basis. By a newmethod, we establish the Loomis-Whitney inequality under the double John basis.As an application, we give another form of the Loomis-whitney inequality.In chapter6, we establish the Busemann inequality about the p-convex body.Using the Busemann inequality, we obtain the γ-convexity of generalization of theradial p convex body and inequalities of intersection body in real and complexspace.