Some Inequalities And Stability In Convex Bodies Geometry

The thesis belongs to the theory of convex geometric analysis, the Brunn-Minkowski theory is the heart of quantitative convexity. As its extension, the L0-Brunn-Minkowski theory, Lp-Brunn-Minkowski theory (p≠0)and dual Brunn-Minkowski theory are high-speed developing geometry branches during the pastover ten years. This thesis is devoted to the study of L0-Brunn-Minkowski in-equality in L0-Brunn-Minkowski theory, volume inequalities for asymmetric Lp-zonotopes in Lp-Brunn-Minkowski theory (p≠0)and stability of inequalities indual Brunn-Minkowski theory. We obtain many meaning results.The research works of this thesis consists of three parts.(1) We extend the recent result of L0-Brunn-Minkowski inequality for twoorigin-symmetric convex bodies in the plane obtained by B¨ r¨ czky, Lutwak, Yangand Zhang, and prove (m≥2) origin-symmetric convex bodies in the plane.Moreover, relying on the recent results of Schuster and Weberndorfer, volume in-equalities for L0-Minkowski combination of origin-symmetric convex bodies inR and its polar form are established. We also systematically provide some proper-ties of L0-Minkowski combination, normalized L0-mixed volume and Wulff shapedetermined by Borel measure and continuous function in L0-Brunn-Minkowskitheory.(2) We establish volume inequalities for asymmetric Lp-zonotopes and theirpolars in Lp-Brunn-Minkowski theory (=0), along with the characterization ofall extremals. These results extend the well-known volume inequalities for origin-symmetric Lp-zonoids and their polars obtained by Ball, Barthe, Lutwak, Yangand Zhang. Moreover, we present volume inequalities for general Wulff shapesand their polars under some symmetric restrictions.(3) We provide three important stability versions of geometric inequalitiesfrom dual Brunn-Minkowski theory. They are stabilities of dual Minkowski in- equality and dual isoperimetric inequality between dual mixed volume, and sta-bility of Lp-dual Aleksandrov-Fenchel inequality between Lp-dual mixed volume.