The ?-convexity Of Sets And Its Basic Properties

Convex analysis is the basic theory for operation and optimization,Since the concept of convex sets was introduced by Minkowski in 1911,convex sets and functions were widely used in operations research,optimization theory,mathematical economics,calculus of variations and other disciplines.With the requirements of applications,a various of generalized convexities of sets have been proposed.These generalized convexities provided a theoretical tools and particular applicable methods for the optimization theory.The research on classical convex analysis has been basically fulfilled.However,as far as we know,there is lack of a systematical treatment on the basic properties of these generalized convexities.So,in this paper,we first collect and analyze some known results about the near convexity,then present a kind of generalized convexity and investigate its basic properties.The main content of this paper includes the following three aspects:In Chapter II,we discuss the relations between the classical convexity and the near convexity simply by their definitions and basic properties.The definition and basic properties of the set TS related to nearly convex sets are also provided.In Chapter III,firstly,inspired by the property of set TS,we introduce the concept of convexity-indicating sets.Secondly,we define the ?-convexity of sets,which is expected as an interpolation between the classical convexity and the near convexity.Some particular nearly convex sets are studied in terms of this new convexity.Then,we explore the basic properties of the ?-convex hull of sets.Finally,some ?-convexity-preserving compositions and operators are discussed.In Chapter IV,we show the relations between nearly convex sets and ?-convex sets.This chapter is the main part of this paper,in which,besides the main contributions,a summary of the thesis is provided,and p some unsolved problems are presented for further studies.Chapter III and IV contain the main contents of this thesis,in particular,Section3.2 and Chapter IV are the highlights.