  We observe that non-doubling metric spaces can be characterized as those that
contain arbitrarily large sets of approximately equidistant points and use this
to show that, for $\gamma \in (0,2]$, the $\gamma$-Liouville quantum gravity
metric is almost surely not doubling and thus cannot be quasisymmetrically
embedded into any finite-dimensional Euclidean space. This generalizes the
corresponding result of Troscheit for the Brownian map (which is equivalent to
the case $\gamma = \sqrt{8/3}$).
