Funk and Hilbert geometries in spaces of constant curvature

International audienceWe develop the Funk and Hilbert geometries of open convex sets of the sphere Sn and of the hyperbolic space Hn. This is done in analogy with the classical theory in Euclidean space. It is rather unexpected that the Funk geometry, whose definition and development use the affine structure of Euclidean space, has analogues in the non-linear spaces Sn and Hn, where there are no homotheties. The existence of a Funk geometry in these spaces is based on some non-Euclidean trigonometric formulae which display some kind of similarity between (the hyperbolic sine of the lengths of) sides of right triangles. The Hilbert metric in each of the constant curvature settings is a symmetrization of the Funk metric. We show that the Hilbert metric of a convex subset in a space of constant curvature can also be defined using a notion of a cross ratio which is proper to that space