Horofunction extension and metric compactifications

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 Gromovcompactifiable 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.