2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/114010Several authors have pointed out the connection between Barbilian's metric introduced in 1934 and the recent study of Apollonian metrics. We provide examples of various distances that can be obtained by Barbilian's metrization procedure and we discuss the relation between this metrization procedure and important Riemannian and generalized Lagrangian metrics. Then we prove an extension of Barbilian's metrization procedure.To appear in Houston J. MathDifferential GeometryMetric GeometryPrimary 53B40; Secondary 53C60, 51K05, 30C62text