Birkhoff formula for conformal compressions of symmetric cones

In this paper we derive the Birkhoff formula for conformal compressions on symmetric cones. Let V be a Euclidean Jordan algebra and let Ω be the associated symmetric cone. Then Ω admits a natural Finsler metric contracted by any conformal compressions of Ω. We show that the Lipschitz constant of a conformal compression of Ω is equal to the hyperbolic tangent of one fourth of the diameter of the image. This is the same relation which was obtained by Birkhoff on positive reals and by Liverani and Wojtkowski on the space of positive definite real matrices.