Horofunction compactifications of symmetric cones under Finsler distances

TL;DR

This paper explores the horofunction compactifications of symmetric cones under Finsler distances, establishing a correspondence with tangent space compactifications and providing explicit characterizations for Thompson and Hilbert distances.

Contribution

It introduces a novel correspondence between horofunction compactifications of symmetric cones and tangent space normed structures, with explicit descriptions and extensions of the exponential map.

Findings

Complete characterization of Thompson distance horofunctions

Explicit extension of the exponential map for symmetric cones

Geometric description of horofunction compactifications in terms of facial structure

Abstract

In this paper we consider symmetric cones as symmetric spaces equipped with invariant Finsler distances, namely the Thompson distance and the Hilbert distance. We establish a correspondence between the horofunction compactification of a symmetric cone $A_+^\circ$ under these invariant Finsler distances and the horofunction compactification of the normed space in the tangent bundle. More precisely, for the Thompson distance on $A^\circ_+$ we show that the exponential map extends as a homeomorphism between the horofunction compactification of the normed space in the tangent bundle, which is a JB-algebra, and the horofunction compactification of $A_+^\circ$. We give a complete characteristation of the Thompson distance horofunctions and provide an explicit extension of the exponential map. Analogues results are established for the Hilbert distance on the projective cone $PA_+^\circ$. The analysis yields a geometric description of the horofunction compactifications of these spaces in terms of the facial structure of the closed unit ball of the dual norm of the norm in the tangent space.