K-convexity Of Convex Sets On Banach Spaces And Its Applications

The convexity study of Banach space is one of the important research contents of Banach space geometry theory.The study of Banach space geometry theory starts from the convexity of the unit ball in Banach space,but the research on some properties of some convexity is not very perfect,so it is of great significance to further study the convexity of Banach space.The various k-convexities of Banach space have a common feature,that is,the unit sphere of Banach space is used as the research object.This paper breaks through the constraint of the unit ball and extends the convexity theory of Banach space to the convex set that is not empty interior.The definitions of k-strictly convex set and k-uniformly convex set are given,and some good properties of k-strictly convex set,k-uniformly convex set,compact midpoint locally uniformly convex space and k-uniformly convex space are obtained.The geometric theory of Banach space is an important part of functional analysis,which not only promotes the development of functional analysis,but also affects the optimization problem such as nonlinear programming and optimal best approximation.As we all know,the excellent special features of convex optimization solve the optimization problem better.Therefore,the Banach space geometry theory is applied to convex optimization.For a specific nonlinear programming problem,after some abstract processing,it can always be transformed into a convex set constraint optimization problem.Using the distance function,the geometric method is used to deal with the problem,and then the specific optimization conditions are obtained.