  A weak metric on a set is a function that satisfies the axioms of a metric
except the symmetry and the separation axioms. In the present paper we
introduced a weak metric, called the Apollonian weak metric, on any subset of a
Euclidean space which is either bounded or whose boundary is unbounded. We then
relate this weak metric to some familiar metrics such as the Poincare metric,
the Klein-Hilbert metric, Funk metric, and the part metric which play an
important role in classic and recent work on geometric function theory.
