Behaviour of distance functions in Hilbert–Finsler geometry

Smooth bounded convex domains equipped with their Hilbert metric provide nice examples of constant negatively curved Finsler manifolds. An important property of these models is that contrary to what happens in Riemannian setting the distance between two points moving at unit speed along intersecting geodesics is not necessarily convex. However we give sharp estimates of the asymptotic behaviour of such functions.