Convex metric space

Definition
A metric space $$(X,d)$$ is said to be convex if given any two distinct points $$x,y \in X$$, there exists a third distinct point $$z \in X$$ such that $$z$$ is between $$x$$ and $$y$$, in the following sense:

$$d(x,z) + d(z,y) = d(x,y)$$