Weak metrics on Euclidean domains

Athanase Papadopoulos (1) , Marc Troyanov (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Section de Mathematiques [Lausanne]

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 Apol lonian 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 Poincar?e 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

Mots clés: 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.; half-Apollonian semi-metric

Domaines: Topologie géométrique [math.GT]; Géométrie métrique [math.MG]

Athanase Papadopoulos, Marc Troyanov. Weak metrics on Euclidean domains. JP Journal of Geometry and Topology, 2007, 7 (1), pp.3-44. ⟨hal-00141702⟩