On orthogonal chords in normed planes
A result due to \textit{E. Makai, jun.} and \textit{H. Martini} [Geom. Dedicata 34, No.~2, 199--209 (1990; Zbl 0688.52002)] shows that in the Euclidean plane a convex body of diameter 1 is of constant width 1 if and only if any two perpendicular intersecting chords of it have total length greater than or equal to 1. The purpose of this paper is to investigate a similar result in the setting of a Minkowski plane \(X\), i.e., a real two-dimensional normed linear space, where the concept of Birkhoff orthogonality is considered: given \(x,y\in X,\) \(x\) is said to be Birkhoff orthogonal to \(y\) (\(x\perp_B y\)) if \(\|x+\lambda y\|\geq \|x\|\) for every \(\lambda\in {\mathbb R}.\) In Section 3, the authors provide examples proving that if one considers Birkhoff orthogonality the previous result cannot be extended to Minkowski planes. In Section 4, the authors present further results on intersecting orthogonal chords of unit circle of a Minkowski plane, where orthogonality is meant in the sense of Birkhoff and in the sense of James.
scientific article; zbMATH DE number 5858877