Horofunction compactifications and duality

TL;DR

This paper explores the topology of horofunction compactifications in Finsler manifolds, establishing homeomorphisms to dual norm balls and analyzing boundary structures in various geometric spaces.

Contribution

It demonstrates that for certain Finsler spaces, the horofunction compactification is homeomorphic to the dual norm ball, providing explicit constructions and analyzing boundary partitions.

Findings

Homeomorphism between horofunction compactification and dual norm ball

Partition of horofunction boundary into parts corresponding to dual faces

Confirmation of duality connection in Euclidean Jordan algebras

Abstract

We study the global topology of the horofunction compactification of smooth manifolds with a Finsler distance. The main goal is to show, for certain classes of these spaces, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the norm in the tangent space (at the base point) that generates the Finsler distance. We construct explicit homeomorphisms for a variety of spaces in three settings: bounded convex domains in $\mathbb{C}^n$ with the Kobayashi distance, Hilbert geometries, and finite dimensional normed spaces. For the spaces under consideration, the horofunction boundary has an intrinsic partition into so called parts. The natural connection with the dual norm arises through the fact that the homeomorphism maps each part in the horofunction boundary onto the relative interior of a boundary face of the dual unit ball. For normed spaces the connection between the global topology of the horofunction boundary and the dual norm was suggested by Kapovich and Leeb. We confirm this connection for Euclidean Jordan algebras equipped with the spectral norm.