  For $\gamma \in (0,2)$, we define a weak $\gamma$-Liouville quantum gravity
(LQG) metric to be a function $h\mapsto D_h$ which takes in an instance of the
planar Gaussian free field (GFF) and outputs a metric on the plane satisfying a
certain list of natural axioms. We show that these axioms are satisfied for any
subsequential limits of Liouville first passage percolation. Such subsequential
limits were proven to exist by Ding-Dub\'edat-Dunlap-Falconet (2019). It is
also known that these axioms are satisfied for the $\sqrt{8/3}$-LQG metric
constructed by Miller and Sheffield (2013-2016).
  For any weak $\gamma$-LQG metric, we obtain moment bounds for diameters of
sets as well as point-to-point, set-to-set, and point-to-set distances. We also
show that any such metric is locally bi-H\"older continuous with respect to the
Euclidean metric and compute the optimal H\"older exponents in both directions.
Finally, we show that LQG geodesics cannot spend a long time near a straight
line or the boundary of a metric ball. These results are used in subsequent
work by Gwynne and Miller which proves that the weak $\gamma$-LQG metric is
unique for each $\gamma \in (0,2)$, which in turn gives the uniqueness of the
subsequential limit of Liouville first passage percolation. However, most of
our results are new even in the special case when $\gamma=\sqrt{8/3}$.
