Extendability of large-scale Lipschitz maps

Let X, Y be metric spaces, S a subset of X, and f:S --> Y a large-scale lip schitz map. It is shown that f possesses a large-scale lipschitz extension (f) over bar (f) over bar: X --> Y (with possibly larger constants) if Y is a Gromov hyperbolic geodesic space or the cartesian product of finitely ma ny such spaces. No extension exists, in general, if Y is an infinite-dimens ional Hilbert space. A necessary and sufficient condition for the extendabi lity of a lipschitz map f : S --> Y is given in the case when X is separabl e and Y is a proper, convex geodesic space.