  It is known that the $\tilde j$-metric, half-apollonian metric and
scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are
defined as a supremum of one-point metrics (i.e., metrics constructed using one
boundary point) and the supremum is taken over all boundary points. The aim of
this paper is to show that taking the average instead of the supremum yields a
metric that preserves the Gromov hyperbolicity. Moreover, we show that the
Gromov hyperbolicity constant of the resulting metric does not depend on the
number of metrics used in taking the average. We also provide an example to
show that the average of Gromov hyperbolic metrics is not, in general, Gromov
hyperbolic.
