The goal of this paper is to introduce and to study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of the hyperbolic space Hn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb H ^n$$\end{document} and of the sphere Sn\documentclass[12pt]{minimal}
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\begin{document}$$S^n$$\end{document}. We highlight some striking similarities among the three cases (Euclidean, spherical and hyperbolic) which hold at least at a formal level. The proofs of the basic properties of the classical Funk metric on subsets of Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb R ^n$$\end{document} use similarity properties of Euclidean triangles which of course do not hold in the non-Euclidean cases. Transforming the side lengths of triangles using hyperbolic and circular functions and using some non-Euclidean trigonometric formulae, the Euclidean similarity techniques are transported into the non-Euclidean worlds. We start by giving three representations of the Funk metric in each of the non-Euclidean cases, which parallel known representations for the Euclidean case. The non-Euclidean Funk metrics are shown to be Finslerian, and the associated Finsler norms are described. We then study their geodesics. The Hilbert geometry of convex sets in the non-Euclidean constant curvature spaces Sn\documentclass[12pt]{minimal}
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\begin{document}$$S^n$$\end{document} and Hn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb H ^n$$\end{document} is then developed by using the properties of the Funk metric and by introducing a non-Euclidean cross ratio. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. This leads to a formulation and a discussion of Hilbert’s Problem IV in the non-Euclidean settings. Projection maps between the spaces Rn,Hn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb R ^n, \mathbb H ^n$$\end{document} and the upper hemisphere establish equivalences between the Hilbert geometries of convex sets in the three spaces of constant curvature, but such an equivalence does not hold for Funk geometries.
