Polyhedral Horofunction Compactification as Polyhedral Ball

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.