Let $(X,d,m)$ be a metric measure space. We say that it is *doubling in the sense of metric spaces* if for every:
$x\in X$ and every $r>0$ there exists some **(metric) doubling constant** $C_d\geq 0$ such that
$$
Ball(x,r) \mbox{ can be covered by at-most $C_d$ balls of radius $r/2$}.
$$

There is a different, related and in some spirit "equivalent", notion of *doubling in the sense of metric measure spaces*, which states that there is a constant $C_m\geq 0$ such that: for every $x\in X$ and each $r>0$
$$
m(Ball(x,r)) \leq C_m m(Ball(x,r/2)).
$$

If $(X,d,m)$ is doubling in the sense of metric measure spaces, with constant $C_m$, then is it doubling in the sense of metric spaces? And if so, can we $C_m$ to deduce an upper-bound for $C_d$?

Note, I'm most interested in the case where $m$ is an $s$-dimensional Hausdorff measure.