Timelike Hilbert and Funk geometries

The Funk and Hilbert geometries for spaces of constant curvature

The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets Ω of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representa...

Isometries of polyhedral Hilbert geometries

We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an n-simplex with n ≥ 2. Moreover, we determine the isometry group of the Hilbert geometry on the n-simplex for all n ≥ 2, and find that it has the collineation group as an index-two subgroup. These results confirm, for the class of pol...

On Rough-isometry Classes of Hilbert Geometries

We prove that Hilbert geometries on uniformly convex Euclidean domains with C 2-boundaries are roughly isometric to the real hyperbolic spaces of corresponding dimension.

Topological Dynamics of Nonstrictly Convex Hilbert Geometries

Crofton Measures in Polytopal Hilbert Geometries

The Hilbert geometry in an open bounded convex set in R is a classical example of a projective Finsler space. We construct explicitly a positive measure on the space of lines in a polytopal Hilbert geometry which yields an integral geometric representation of Crofton type for the Holmes-Thompson area of hypersurfaces. MSC 2000: 53C60 (primary); 53C65, 52B11 (secondary)