On some Finsler structures of symmetric cones

For a simple Euclidean Jordan algebra, it turns out that the corresponding symmetric cone Omega 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 Omega can be viewed as "minimal geodesic curves" for the Finsler metric and that the exponential mapping of Omega increases Finsler distances. Furthermore, it is shown that every Finsler ball on Ome ga is convex.