The Applications Of Shadow Systems In Convex Geometric Analysis

The researches of this dissertation belong to the theory of convex geometric analysis, of which the core part is the Brunn-Minkowski theory. Rogers and Shephard( [125, 126]) firstly proposed the definition of shadow systems. This dissertation is devoted to the a?ne isoperimetric inequalities and the related extremal value problems in convex geometric analysis by using the tool of shadow systems.In chapter 2, we first give the definitions of the Orlicz mean zonoids of convex bodies, and show that the Orlicz mean zonoids of convex bodies are convex bodies.Then, we discuss the continuity of the Orlicz mean zonoid operator, and obtain that the volume ratios of the convex bodies and their Orlicz mean zonoids are a?ne invariant. At the end of this chapter, for convex bodies, we get the extremal values of the volume ratios of their Orlicz mean zonoids and themselves. The definitions and main results in this chapter generalize the ones of the professor Zhang Gaoyong’s paper partly in Geom. Dedicata in 1991.In 1998, professor Zhang Gaoyong established the radial mean bodies on his paper in Amer. J. Math., and obtained the famous Rogers-Shephard inequality and the inverse form of the well-known Petty projection inequality by this notation. In chapter 3, by the dual properties of convex bodies, we give the definition of the Orlicz mean body HφK of a convex body K, which is also a convex body.Some basic properties about the Orlicz mean bodies are gotten. Then, the a?ne isoperimetric inequalities involving the volumes of the convex bodies and their Orlicz mean bodies are obtained. The processes of the main results also reflect the tool of the shadow system useful and powerful.In Chapter 4, we continue the research on the Orlicz mean zonoid operator,and discuss the applications of the Orlicz mean zonoid operator as the operator of two variables. The ideas and techniques of Gardner, Hug and Weil [39],Pfiefer [121], Zhang [162] and Lutwak, Yang and Zhang [98] play a critical role throughout this chapter. We also establish an a?ne isoperimetric inequality for the Orlicz mean zonoid operator.In chapter 5, we establish the Orlicz-Brunn-Minkowski inequalities for polar bodies and dual star bodies. These results can be considered as polar counterparts of the existing Orlicz-Brunn-Minkowski inequality and its dual.