2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132115We show that a geodesic metric space is hyperbolic in the sense of Gromov if and only if intersections of balls have bounded eccentricity. In particular, $\R$-trees are characterized among geodesic metric spaces by the property that the intersection of any two balls is always a ball. Both Gromov hyperbolicity and CAT($κ$) geometry can be characterised in terms of the geometry of the intersection of balls.Group Theory20F67; 20F65; 20E08text