Horofunction Extension and Metric Compactification

Abstract

A necessary and sufficient condition for the horofunction extension (X,d)h of a metric space (X,d) to be a compactification is hereby established. The condition clarifies previous results on proper metric spaces and geodesic spaces and yields the following characterization: a Banach space is Gromov-compactifiable under any renorming if and only if it does not contain an isomorphic copy of ℓ1. In addition, it is shown that, up to an adequate renorming, every Banach space is Gromov-compactifiable. Therefore, the property of being Gromov-compactifiable is not invariant under bi-Lipschitz equivalence.