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. The aim of this paper is to present some interesting weak metrics and to study some of their properties. In particular, we introduce 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 relate this weak metric to some familiar metrics such as the Poincaré metric, the Klein-Hilbert metric, the Funk metric and the part metric which all play important roles in classical and in recent work on geometric function theory.
Keywords: weak metric ; Apollonian weak metric ; hyperbolic geometry ; Poincaré model ; Klein model ; Klein-Hilbert metric ; Funk weak ; metric ; part metric ; Poincaré metric ; Apollonian ; semi-metric ; half-Apollonian semi-metric.
Record created on 2008-03-16, modified on 2016-08-08