We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction … compactification with respect to a polyhedral Finsler metric.
In this paper we answer positively a question raised by Kapovich and Leeb in a recent paper titled "Finsler bordifications of symmetric and certain locally symmetric spaces". Specifically, … we show that for a finite-dimensional vector space with polyhedral norm, its horofunction compactification is homeomorphic to the dual unit ball of the norm by an explicit map. To prove this we establish a criterion for converging sequences in the horofunction compactification, and generalize the basic notion of the moment map in the theory of toric varieties.