Straight projective-metric spaces with centres (Q1754410)

Let \(({\mathcal M},d)\) be a metric space such that \(\mathcal M\) is an open convex subset of \({\mathbb R}^n\), the metric is complete and induces the standard topology on \(\mathcal M\), and the geodesics of \(\mathcal M\) are precisely the intersections of straight lines of \({\mathbb R}^n\) with \(\mathcal M\). If every point of \(\mathcal M\) is the centre of a metric reflection, then \(\mathcal M\) is either isometric to hyperbolic space, or \(\mathcal M\) equals \({\mathbb R}^n\) and \(d\) is a Minkowski metric. For the theorem to hold, it is actually sufficient to assume that there exist metric reflections for finitely many centres whose distances satisfy certain suitably chosen irrationality assumptions.