  This paper provides rates of convergence for empirical (generalised)
barycenters on compact geodesic metric spaces under general conditions using
empirical processes techniques. Our main assumption is termed a variance
inequality and provides a strong connection between usual assumptions in the
field of empirical processes and central concepts of metric geometry. We study
the validity of variance inequalities in spaces of non-positive and
non-negative Aleksandrov curvature. In this last scenario, we show that
variance inequalities hold provided geodesics, emanating from a barycenter, can
be extended by a constant factor. We also relate variance inequalities to
strong geodesic convexity. While not restricted to this setting, our results
are largely discussed in the context of the $2$-Wasserstein space.
