The least doubling constant of a metric measure space

ABSTRACT: Given a metric measure space $(X,d,\mu)$, its doubling constant is given by
$$
C_\mu=\sup_{x\in X, r>0} \frac{\mu(B(x,2r))}{\mu(B(x,r))},
$$
where $B(x,r)$ denotes the open ball of radius $r$ centered at $x$. Clearly, $C_\mu\geq1$, and in the case $X$ reduces to a singleton $C_\mu=1$. One might think that for a metric space with more than one point, the constant $C_\mu$ could be very close to one. However, we will show that in general $C_\mu\geq2$. The talk is based on a joint work with Javier Soria (Barcelona).