  We quickly review and make some comments on the concept of convexity in
metric spaces due to Takahashi. 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 discuss
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. Finally, we apply some
of our results to the metric projection problem and fixed point theory.
