Horofunction Compactifications and Duality

Abstract

We study the global topology and geometry of the horofunction compactification of classes of symmetric spaces under Finsler distances in three settings: bounded symmetric domains of the form B=B_1times cdots times B_r, where B_i is an open Euclidean ball in {mathbb {C}}^{n_i}, with the Kobayashi distance, symmetric cones with the Hilbert distance, and Euclidean Jordan algebras with the spectral norm. For these spaces we show, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the Finsler metric in the tangent space at the basepoint. In each case we give an explicit homeomorphism. For finite dimensional normed spaces the link between the geometry of the horofunction compactification and the dual unit ball was suggested by Kapovich and Leeb, which we confirm for Euclidean Jordan algebras with the spectral norm. Our results also show that this duality phenomenon not only occurs in normed spaces, but also in a variety of noncompact type symmetric spaces with invariant Finsler metrics.