The least doubling constant of a metric measure space

We study the least doubling constant $C_{(X,d)}$, among all doubling measures $\mu$ supported on a metric space $(X,d)$. In particular, we prove that for every metric space with more than one point, $C_{(X,d)}\ge 2$. We also describe some further properties of $C_{(X,d)}$ and compute its value for several important examples.