Characterization Of Valuations And Its Applications

This dissertation deals with topics valuations theory in Convex Geometry, which is a high-speed developing geometry branch during the past over ten years. This thesis is devoted to characterizations of geometric operators and its related topic.The reseach works of this thesis consists of five parts.The radial Blaschke-Minkowski homomorphism is a radial Minkowski valuation which is more general than the intersection body operator. Schuster characterized completely all radial Blaschke Minkowski homomorphisms, and studied the Busemann-Petty type prob-lem for the radial Blaschke-Minkowski homomorphism. In chapter 2, We focus on the study of the Busemann-Petty type problem for mixed radial Blaschke-Minkowski homo-morphisms. We generalized Schuster’s results.Lutwak, Yang, and Zhang introduced the Lp centroid body Γ-PK of the convex body K. In chapter 3, we introduced the Lp-mixed centroid body Γ_viK, and obtained the Fourier analytic formula for Lp-mixed centroid body. Moreover, we studied the Busemann-Petty type problem for the Lp-mixed centroid body.Abardia and Bernig introduced the complex projection body, obtained the charac-terization theorem of the complex projection body operator, and established the general Minkowski inequality, Aleksandrov-Fenchel inequality and Brunn-Minkowski inequality for mixed complex projection bodies. In chapter 4, we established the general Minkowski inequality, Aleksandrov-Fenchel inequality and Brunn-Minkowski inequality for polars of mixed complex projection bodies, which are polar forms of their’s results.Zhu, Zhou, and Xu defined an Orlicz radial sum and dual Orlicz mixed volumes for convex functions, and established the dual Orlicz-Brunn-Minkowski inequality for convex functions. In chapter 5, we first defined an Orlicz radial sum and dual Orlicz mixed volumes for concave functions, and then established the dual Orlicz-Brunn-Minkowski inequality for concave functions.Boroczky, Lutwak, Yang, and Zhang established the log-Brunn-Minkowski inequality and the log-Minkowski inequality for origin-symmetric convex bodies in the plane. For n> 3, Boroczky, Lutwak, Yang, and Zhang conjectured that there exists the log-Brunn-Minkowski inequality and log-Minkowski inequality for origin-symmetric convex bodies in Rn. In chapter 6, we define the log radial sum for star bodies, establish the dual log-Brunn-Minkowski inequality and the dual log-Minkowski inequality. Moreover, the equivalence between the dual log-Brunn-Minkowski inequality and the dual log-Minkowski inequality is demonstrated.