Halving circular arcs in normed planes (Q1046750)

Let \(S\) be the unit circle of a Minkowski plane \((X, \| \;\,\| )\) with the fixed orientation and with the (Minkowskian) length \(| S| \). For \(p,q\) in \(S\), let \(\delta(p,q)\) be the length of the part of \(S\) connecting \(p\) to \(q\) in the positive orientation. For given \(x\in S\) and \(\alpha\in (0,1)\), let \(x_{\alpha}\) and \(x_{\alpha}^{-1}\) be points of \(S\) such that \(\delta(x_{\alpha}^{-1},x)=\delta(x,x_{\alpha})=\alpha| S| \). The main result of the paper: The plane \(X\) is Euclidean if and only if \(\| x-x_{\alpha}\| =\| x- x_{\alpha}^{-1}\| \) for any \(x\in S\) and any irrational \(\alpha\in (0,1)\).