Timelike Hilbert geometry of the spherical simplex

Abstract

We prove the following result on the timelike spherical Hilbert geometry of simplices: Let ${\Delta }_2$ be a simplex on the 2-sphere and ${\tilde{\Delta }}_2$ the antipodal simplex. We show that the timelike spherical Hilbert geometry associated with the pair ${\Delta }_2,{\tilde{\Delta }}_2$ is isometric to a union of six copies of vector spaces equipped with a timelike norm, isometrically and transitively acted upon by the group ${\mathbb{R}}_{>0}^2\times {\mathbb{Z}}_3\times {\mathbb{Z}}_2$. This is a timelike spherical analogue of a well-known result (due to Busemann) stating that the Hilbert metric of a Euclidean simplex is isometric to a metric induced by a normed vector space. At the same time, this gives a new example of timelike space.