On Basic Properties Of Several Convexities Of Sets In Finite Dimension Spaces

This thesis aims at dealing with several important convexities of sets in finitedimensional spaces, including near convexity, close convexity and so on. Since, in order toprovide basic tools in solving optimal problems, the concept of convexity for sets wasintroduced by Minkowski in1911, various kinds of (weaker) convexity were introduced bymathematicians, and all of these weaker convexities have been successfully used inoptimization theory. However, as far as we know, there seems no systematic investigationson the basic properties of these convexities except the classical one. So, in our work, wetry first to collect some known important results and known approaches used in dealingwith these convexities. Then we try to make some contributions to the properties of theseconvexities. This thesis is divided into five chapters. The first two chapters are introductionand preliminaries, while our main work is contained in the last three chapters.In Chapter III, we recalled first the definition of nearly convex sets, then we studiedproperties of the set determined by a nearly convex set, from which we deduce someproperties of nearly convex sets. Some examples are also given to show the differencesbetween convex sets and nearly convex sets. Finally, we defined the ɑ-nearly convex hullof a set and then studied its properties.In Chapter IV, we recalled first the definition of closely convex sets and then showsome properties of closely convex sets, discussing, through arguments or examples,whether the sum, product, affine image and intersection of some closely convex set(s) arestill closely convex.In Chapter V, we recalled first the definitions of three conic convexities and thenprovide some research in the relations between these conic convexities. Finally we try toimprove some known results, by weakening conditions, on these conic convexities.