Horofunction Extension of Metric Spaces and Banach Spaces

Explore a 17-minute conference talk by Sebastián Tapia García from the Workshop on "Structures in Banach Spaces" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in March 2025. Discover a necessary and sufficient condition for the horofunction extension of a metric space to be a compactification. Learn how this condition clarifies previous results on proper metric spaces and geodesic spaces, leading to an important characterization: a Banach space is Gromov-compactifiable under any renorming if and only if it does not contain an isomorphic copy of ℓ1. Understand how, with adequate renorming, every Banach space can be Gromov-compactifiable, demonstrating that this property is not invariant under bi-Lipschitz equivalence. The research presented is a collaborative effort with A. Daniilidis, M.I. Garrido, and J. Jaramillo.