Some Properties Of Spherical Convex Bodies In S^n-1

There are many different definitions of spherical convex set.In this article,we adopt a pure analytic approach to defining the strong convexity for sets inn-1S and investigating some basic properties of spherically convex sets,which makes it possible to formulate and demonstrate some conclusions that can hardly be done in geometric methods.The main achievements of this paper is:(1)We present a pure analytic approach to defining the convex sets inn-1S;(2)We give the properties of s-convex sets and s-convex hull of sets;(3)We establish a structure theorem of Minkowski type for closed s-convex sets;(4)We give some s-convexity-preserving operations?The full text is divided into five chapters.Chapter I presents a brief survey of the research in this subject and our main work.In Chapter II,we collect some important definitions,properties and theorems of convex set innR.In Chapter III,we define the spherical convex set in an analytic method and show some elementary properties of the spherical convex sets.Chapter IV focus on the s-convexity-preserving operations.At the end,there is a summary of this thesis where the future considerations are mentioned at the same time.