Horofunction compactifications of symmetric cones under Finsler distances

Abstract

In this paper we consider symmetric cones equipped with invariant Finsler distances, namely the Thompson distance and the Hilbert distance. We give a complete characterisation of the horofunctions of the symmetric cone \(A_+^\circ\) under the Thompson distance and establish a correspondence between the horofunction compactification of \(A_+^\circ\) and the horofunction compactification of the normed space in the tangent bundle. More precisely, 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 \(A_+^\circ\). Analogues results are established for the Hilbert distance on the projective symmetric cone \(PA_+^\circ\). The analysis yields a concrete 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.