We explain why a geodesic metric space is delta-hyperbolic in the sense of Gromov if and only if the intersection of any two metric balls is almost a prescibed ball. In particular, R-trees can be characterised by the property that the intersection of any two metric balls is exactly a particular ball.
This is joint brainstorming with Graham Niblo. An earlier version of the paper had mistakes, and the current correct version has several unsatisfactory points that I would like to run to the audience. This talk should be very accessible.