Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
We then say that a set $S\subseteq X$ is convex if for all $x,y \in S$ it holds true that $\left [ x,y \right ] \subseteq S$. Denote by $\tau$ the topology on $X$ induced by the metric $d$.
My question is does there exist a family $\mathcal{B}\subseteq \tau$ of convex sets such that $\mathcal{B}$ is a basis for the topology $\tau$?
It should be noted that open and closed balls in metric spaces are not necessarily convex sets. Also, arbitrary intersection of convex sets in metric spaces is a convex set. Thus, we can define convex hulls.