Valuation Theory And Related Topics

The thesis belongs to convex geometrical analysis theory, which is a high-speed devel-oping geometry branch during the past several decades years. This thesis is devoted to the study of classificaiton problems of convex body valued valuations of valuation theory, and to the study of some geometrical objects on Orlicz space. These problems have attracted increased interest for this direction, which refer to classificaiton of Lp-Blaschke valuations, Orlicz geominimal surface, mixed Orlicz quermassintegrals, asymmetric Orlicz zonotopes and equivalence properties of inequalities.Convex body valued valutions consists of Minkowski valution and Blaschke valuation, by the difference of addition operation. In2011, C. Haberl established a classification of all continuous symmetric Blaschke valuations, which are compatible with the general lin-ear group. Chapter2extends Haberl’s results in the context of the Lp-Brunn-Minkowski theory when p>1for n≥2, and establishs a classification of continuous, linearly inter-twining, symmetric Lp-Blaschke valuations. We apply the ideas of normalization which is gived by Lutwak,Yang and Zhang, and get the characterization of normalized Lp-Blaschke valuations, and show that for dimensions n≥3, the only continuous, linearly intertwin-ing, normalized symmetric Lp-Blaschke valuation is the normalized Lp-curvature image operator.In Chapter3, we introduce the definition of asymmetric Orlicz zonotopes Z+Φ and shadow systems are used to give a sharp upper estimate for volume of Z+Φ and a sharp lower estimate for volume of the polar of Z+Φ in terms of the volume of Z+Φ.In1996, E. Lutwak extended the important concept of geominimal surface area to Lp version, which serves as a bridge connecting a number of areas of geometry:affine dif-ferential geometry, relative differential geometry, and Minkowskian geometry. In Chapter4, by using the concept of Orlicz mixed volume, we extend geominimal surface area to the Orlicz version and give some properties and an isoperimetric inequalities for the Orlicz geominimal surface areas.In Chapter5, following the work of Gardner, Hug and weil, we introduce the con-cept of mixed Orlicz quermassintegrals, which extend mixed p-quermassintegrals to Orlicz version, and some related inequalities are established. As an application, we study the so-called i-th Orlicz geominimal surface area, which is an extension of Lutwak’s work.Brunn-Minkowski inequality and Minkowski inequality are two important and funda-mental inequalities in Convex Geometric Analysis. Quite recently, Gardner, Hug and Weil established the Orlicz extention of these two inequalities, and constructed a general frame-work for the Orlicz-Brunn-Minkowski theory. The purpose of Chapter6is to point out the equivalence properties of these four inequalities:classical Brunn-Minkowski inequality, classical Minkowski inequality, Orlicz-Brunn-Minkowski inequality and Orlicz-Minkowski inequality.