Basic Properties Of ?-Convex Sets

Convex analysis plays an important role in many mathematics branches,especially in operation and optimization.Since the concept of convex set was introduced by Minkowski in 1911,convex analysis became the fundamental theory of optimization theory,mathematical economics,calculus of variations and other fields.With the requirement of applications,a variety of generalized convexities about sets have been proposed by scholars.The classical convex analysis is closely to be completed,but as far as we know,the research on the fundamental theory of generalized convexities is not much in the literature.For instance,until now it is not known whether or not the well-known Radon's,Caratheodory's,Helly's Theorem and the Minkowski's structure theorem hold for ?-convex sets.So,in this thesis,we are going to deal with these problems.The thesis is divided into four chapters.The first chapter is an introduction to the topics in the thesis,the background and the development in these topics.The second chapter is a collection of notation and known results on convex sets and?-convex sets,including the classical Radon's theorem,.Caratheodory's theorem,Helly's theorem on convex sets and the Minkowski's structure theorem for convex sets;also the basic properties of the so-called ?-convex sets and the T-convex hulls of sets.Chapter ? is devoted to discussing the Radon's,Caratheodory's and Helly's theorem on ?-convex sets.The last chapter focuses on the Minkowski's structure theorem for closed ?-convex sets,where the concepts of T-extreme points and T-faces of closed T-convex sets are introduced...The main contributions of this thesis are in Chapter ? and Chapter ?,where we proved under some suitable conditions the combinatory theorems and structure theorem for ?-convex sets.