  In the first part of this investigation, [Ha], we generalized a weighted
distance function of [Li] and found necessary and sufficient conditions for it
being a metric. In this paper some properties of this so-called M-relative
metric are established. Specifically, isometries, quasiconvexity and local
convexity results are derived. We also illustrate connections between our
approach and generalizations of the hyperbolic metric.
