Research On Some Basic Properties Of Spherically Convex Sets

In this thesis,some spherical analogues of Euclidean convexity theory are studied through collecting and analyzing the known results,the commonly used methods in Euclidean convexity theory.In particular,the combinatorical and structural properties of spherically convex sets are discussed in details.A number of results parallel to those in Euclidean spaces are established in spherical spaces.The achievements obtained in this thesis might provide some helps in establishing a systematic spherical convexity theory.The main contributions of this thesis are:(?)Some properties of s-extreme(exposed)sets of a spherical convex set are given;(?)Several structural theorems of spherical closed convex sets are established;(?)Some transformations,such as the spherical translations,on the sphere are defined;(?)A number of theorems of Helly's type on spherical convex sets related to the above translations are formed.The thesis consists of six chapters.Chapter ? describes the background,the development process,the current situation and the remaining problems in spherical convexity theory,and introduces the main contributions of our work.Chapter ? collects some basic concepts,terms,common symbols and marks,basic definitions,and the known results(stated as lemma,propositions and theorems etc)to be used in the full text.In Chapter ?,we define the s-extreme(s-exposed)faces(points)of an s-convex set and study their basic properties,in particular,we show a theorem of Strasszewicz's type about s-extreme points and s-exposed points.In Chapter ?,several structural theorems of Minkowski's type for spherical closed convex sets are established.Chapter ? is devoted to constructing some transformations,such as spherical translations etc,on the sphere,and establishing several theorems of Helly's theorems for s-convex sets related to the spherical translations.At the end,there is a summary of the work in this paper which refers to the future prospects.