Applications Of Valuations To Lp-Brunn-Minkowski Theroy

The thesis belongs to the Lp-Brunn-Minkowski theory, which is a high-speed devel-oping geometry branch during the past over ten years. This thesis is devoted to the study of applications of valuations to Lp-Brunn-Minkowski theory, especially to Shephard type problem and Busemann-Petty type problem.The reseach works of this thesis consists of three parts.Analogue to the (n-1)-dimensinal section fuction, we show tat the (n-j)-dmsensional sction function is also the log-concave fnction. Then we use it to generalize the Busemann's inequality. As applications, we define a generalization of intersection bodies. Finally, we get the dual Brunn-Minkowski inequality of the generalized intersection bodies.In Lp-Brunn-Minkowski theory, we introduce the concept of the Lp-dual affine surafce areaΩ_p and study it systematically. we also establish the affine Isoperimetric inequality, Blaschke-Santalo inequality and Brunn-Minkowski inequality for it. Ludwig extended the generating function of affine surface area from concave functions to convex functions. Thus, we define the Lp-dual affine surafce areaΩp. Though Lp-dual affine surafce areaΩ_p is upper semicontinuous, Lp-dual affine surafce areaΩp is lower semicontinuous. The affine Isoperimetric inequality, Blaschke-Santalo inequality and dual Brunn-Minkowski inequality for Lp-dual affine surafce areaΩp are presented.As far as we know, the projection body operator and the intersection body operater both define a homogeneous of degree n-1, continuous and SO(n) equivariant valua-tion. Based on this, Schuster introdeced Blaschke-Minkowski homomorphisms and radial Blaschke-Minkowski homomorphisms, and study Shephard type problem and Busemann-Petty type problem for them, respectively. We introduce two kinds of valuation:Lp-Blaschke-Minkowski homomorphisms and Lp-radial Minkowski homomorphisms. By us-ing spherical harmonics, convolution and Legendre polynomials, we classified Lp-radial Minkowski homomorphism and Lp-Blaschke-Minkowski homomorphism with radial func-tion and Lp-surface area measure, respectively. The main emphasis of our research is to study Shephard type problem and Busemann-Petty type problem for them. Previous results by Schuster are generalized to a large class of Lp-valuations.