ar X iv : m at h / 06 09 23 6 v 1 [ m at h . M G ] 8 S ep 2 00 6 WEAK METRICS ON EUCLIDEAN DOMAINS

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 Poincaré 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. AMS Mathematics Subject Classification: 30F45, 51M15, 51K99.