Dual Brunn-minkowski Theory In Real Spaces And Complex Spaces

This thesis devotes to study dual Brunn-Minkowski in real spaces and com-plex spaces. For real spaces, we study the unitized Lp mixed intersection body ofstar body, and obtain many inequalities which are the dual forms of the known re-sults. We also establish an extension of connections between an generalized analyticBusemann-Petty problem and positive defnite distributions. For complex spaces,we introduce two complex radial combinations and use them to study the complexdual Brunn-Minkowski theory.The main results are as follows.(1). By applying the integral methods, we establish some quasi Lp Busemann-Petty inequality for Lp mixed intersection body and its star dual body. Further-more, using the relationship between star dual and the operator Ip, we establishquasi Lp Busemann-Petty intersection inequality for Lp mixed intersection bodyof star dual body.(2). We extend the connection between an generalized analytic Busemann-Pettyproblem and positive defnite distributions. We show that the class of λ intersectionbodies is closely related to the generalized analytic Busemann-Petty problem.(3). We introduce the concepts of complex radial combination and complexradial-Blaschke combination, and obtain the relations between those two combina-tions and dual mixed volumes. Then, we extend the properties of real intersectionbody to the complex case, and prove some complex geometric inequalities aboutcomplex intersection bodies and complex mixed intersection bodies. Moreover, asapplications, we get some corollaries including an isoperimetric type inequality anda uniqueness theorem.