  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.
