The type of minimal branching geodesics defines the norm in a normed space (Q2259295)

The starting point of the paper is that all strictly convex spaces are indistinguishable by the types of minimal curves in the sense that the unique minimal geodesic (shortest route) between two points is always the straight line. The authors first look at polyhedral norms on an \(n\)-dimensional space \(N\) and show that two such norms on \(N\) have the same geodesics if and only if they have the same partitions. Next the authors replace geodesics by something a little more involved in order to distinguish also between strictly convex spaces. Instead of minimal geodesics they look at Fermat points for selections of three points and define two normed spaces to be \(F_3\)-indistinguishable if for any triple the set of their Fermat points are congruent (with respect to the two norms). They now prove (Theorem 1) that any strictly convex norm on \(\mathbb{R}^2\) such that the unit circle is invariant under rotation by \(\pi/3\) is \(F_3\)-indistinguishable from the Euclidean norm. A partial converse (Theorem 2) is that if a norm (written \(r(\varphi\)) in polar coordinates) is \(F_3\)-indistinguishable from the Euclidean morm and is differentiable on the unit sphere except possibly at a finite number of points, then \(r(\varphi)=r(\varphi+\pi/3)\) for all \(\varphi\). The paper ends by comparing the achieved results with a result from \textit{C. Benítez} et al. [Trans. Am. Math. Soc. 354, No. 12, 5027--5038 (2002; Zbl 1030.46009)] to conclude that to \(F_3\)-distinguish Hilbert spaces in 2 dimensions is different than in higher dimensions.