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Halving circular arcs in normed planes. (English) Zbl 1199.46064

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 \(\alpha\in (0,1)\).

MSC:

46C15 Characterizations of Hilbert spaces
46B20 Geometry and structure of normed linear spaces
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
51B20 Minkowski geometries in nonlinear incidence geometry
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References:

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