A characterization of hyperbolic spaces. (Q2472584)

The authors prove the following characterization of Gromov hyperbolicity: A geodesic metric space \((X,d)\) is Gromov hyperbolic if and only if the intersection of any two metric balls is at uniformly bounded Hausdorff distance from a metric ball. The authors also obtain an analogous statement for \(\text{CAT}(\kappa)\) spaces, \(\kappa\leq 0\). The result is stated in terms of the `eccentricity' of a set, defined as follows: Given \(\delta\geq 0\), a subset \(S\) of a metric space \(X\) has eccentricity \(\leq\delta\) if there exists \(R\geq 0\) such that \(B(c,R)\subset S\subset B(c',R+\delta)\) for some \(c,c'\in X\). Here, \(B(c,R)\) denotes the ball of radius \(R\) and center \(c\). The authors also prove that \(\mathbb{R}\)-trees, in the category of metric spaces, are characterized by the fact that the intersection of any two metric balls is a metric ball.