Timelike Hilbert geometry of the spherical simplex

Abstract

We prove the following result on the timelike spherical Hilbert geometry of simplices: Let $Δ_2$ be a simplex on the 2-sphere and $\tildeΔ_2$ the antipodal simplex. We show that the timelike spherical Hilbert geometry associated with the pair $Δ_2, \tildeΔ_2$ 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.