Spherically Convex Sets and Spherically Convex Functions

We define first the spherical convexity of sets and functions on general curved surfaces by an analytic approach. Then we study several kinds of properties of spherically convex sets and functions. Several analogies of the results for convex sets and convex functions on Euclidean spaces are established or rediscovered for spherically convex sets and spherically convex functions, such as the Radon-type, Helly-type, Carathéodory-type and Minkowski-type theorems for spherically convex sets, and the Jensen's inequality for spherically convex functions etc. The results obtained here might have applications in some areas, e.g. in the optimization theory on general spherical spaces.