The General Properties Of Spherically Convex Functions

This thesis aims at studying some basic properties,such as the continuity of spherically convex functions etc.First,two available definitions of different form for spherically convex functions are discussed,and related conclusions for spherically convex functions,including the basic properties and the relations between these two kinds,are collected and analyzed.Then,as main contributions,we discuss the Lipschitz-continuity of a spherically convex function,showing that a spherically convex function is Lipschitzian on each closed s-convex subset contained in the relative interior of its domain.Finally,the equicontinuity of a family of spherically convex functions is investigated,which leads to a useful conclusion that,under some very weak conditions,a family of spherically convex functions is naturally equi-Lipschitian,and as an application of this conclusion,it is deduced that,in some sense,the pointwise convergence of a sequence of spherically convex functions implies its uniform convergence.