On some Finsler structures of symmetric cones

For a simple Euclidean Jordan algebra, it turns out that the corresponding symmetric cone Ω has a natural Riemannian metric and it also admits an invariant Finsler metric. In this paper, we show that the geodesics on the Riemannian symmetric space Ω can be viewed as "minimal geodesic curves" for the Finsler metric and that the exponential mapping of Ω increases Finsler distances. Furthermore, it is shown that every Finsler ball on Ω is convex.