For the abstract convex structure in convex metric spaces， three new generalized W-convex functions are proposed， and the method of using midpoint W-convexity to study generalized convexity in convex metric spaces is proposed. Firstly， the concepts of three generalized convex functions based on standard convex structure in linear space are introduced into convex metric space, and three new generalized W-convex functions are defined； Secondly， under some appropriate conditions， it is proved that the intermediate point W-convex function is a midpoint W-convex function， and the midpoint W-convex function is also ［0,1］∩Q-W-convex function. Furthermore we obtain density theorem and discuss some applications of the density theorem in minimization and multiobjective programming problem；Finally， several discriminant criterions of W-convex function are established under the conditions of midpoint W-convexity and upper semicontinuity or lower semicontinuity or W-quasiconvexity or W-strictquasiconvexity or W-semi-strict quasiconvexity. The method of obtaining density theorem and establishing criterions by using midpoint convexity can be applied to the study of the related problems of other types of convexity or generalized convexity.