Finsler metrics on symmetric cones

Abstract.

Let V be a simple Euclidean Jordan algebra with an associative inner product $\langle \cdot|\cdot \rangle,$ and let $\Omega$ be the corresponding symmetric cone. Let ${\mathcal J}(V)$ be the compact symmetric space of all primitive idempotents of V. We show that the function s(a,b) defined by

\[ s(a,b)=\sup\limits_{c\in{\mathcal J}(V)}|\log\langle a|c\rangle-\log\langle b|c\rangle| \]

is a $G(\Omega)$ (the automorphism group of $\Omega$)-invariant complete metric on $\Omega$ and it coincides with a natural Finsler distance on $\Omega.$ We also show that the metric s(a,b) (strictly) contracts any (strict) conformal compression of $\Omega$.