Hilbert geometries with Riemannian points (Q2309449)

A Hilbert geometry \((\mathcal{M},d)\) consists of an open, strictly convex, and bounded subset \(\mathcal{M}\) of \(\mathbb{R}^n\), \(n\in \mathbb{N}\), and the Hilbert metric \(d\), given via the logarithm of certain cross ratios. The author defines a Riemannian point \(P\) of \((\mathcal{M}, d)\) by requiring that the Finsler norm on the tangent space of \(P\) is quadratic. His main result says that for \(n=2\) a Hilbert geometry \((\mathcal{M}, d)\) with two Riemannian points \(P\) and \(Q\), such that the boundary of \(M\) is twice differentiable at the points of intersection of the line \(\overline{PQ}\), must be a Cayley-Klein model of the hyperbolic plane, the boundary of \(\mathcal{M}\) is an ellipse. The author generalizes this result to arbitrary finite dimension \(n\). A reformulation of the main result in terms of geometric tomography is given.