Convexity In Infinite-dimensional Spaces And The Complex L_p Mixed Petty Projection Inequality

The research content of this thesis belongs to modern convex geometry analysis,including the field of convex analysis and the L_pBrunn-Minkowski theory.It focuses on the topological property of convex sets in the field of convex analysis,and the complex L_pcase of mixed projection bodies in L_pBrunn-Minkowski theory,including the property of the relative interior,the relative boundary and the closure of convex sets in the framework of infinite dimensional linear spaces;the general complex L_pmixed projection bodies,their properties and related important inequalities.The specific research content is as follows:1.Under the framework of infinite dimensional linear spaces(i.e.general linear topological spaces),it does not necessarily hold that the relative interior of convex sets is non-empty and the affine hull of the closed set is still a closed set.Under the condition that the affine hull of the closure of a convex set is equal to its affine hull,a purely geometric representation of the topological concept of a convex set in infinite dimensional linear spaces is given.By the equivalent definition of the affine hull,the equivalent description of the condition that the affine hull of the closure of a convex set closure is equal to its affine packet is obtained,and the topological property of convex sets is further discussed.2.Recently,Wang and Liu first proposed the concept of complex L_pprojection bodies and obtained Petty projection inequality for complex L_pprojection bodies by combining the property of asymmetric L_pzonoids.As a general case of complex L_pprojection body,we introduce the concept of complex L_pmixed projection bodies and prove the general quermassintegral form of Petty projection inequality for complex L_pmixed projection bodies.