A convexity of functions on convex metric spaces of Takahashi and applications

We show that Takahashi's idea of convex structures on metric spaces is a natural generalization of convexity in normed linear spaces and Euclidean spaces in particular. Then we introduce a concept of convex structure based convexity to functions on these spaces and refer to it as W-convexity. W-convex functions generalize convex functions on linear spaces. We provide illustrative examples of (strict) W-convex functions and dedicate the major part of this paper to proving a variety of properties that make them fit in very well with the classical theory of convex analysis. As expected, the lack of linearity forced us to make some compromises in terms of conditions on either the metric or the convex structure. Finally, we apply some of our results to the metric projection problem and fixed point theory.