2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/158047A 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.Metric Geometry30F45; 51M15; 51K99text