2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128574We provide examples of non-locally compact geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.11 pages, 2 figuresMetric Geometry53C20text