A characterization of hyperbolic spaces
  We 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($\kappa$) geometry can be characterised in terms of the geometry of the intersection of balls. 
Indira Chatterji and Graham A. Niblo