Convexity and Geodesic Metric Spaces

In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metric spaces.