Halving circular arcs in normed planes
type: Item
instance of: scholarly article
title: Halving circular arcs in normed planes
zbMATH Open document ID: 1199.46064
author: Horst Martini
author: Senlin Wu
published in: Periodica Mathematica Hungarica
publication date: 2009-12-28T00:00:00Z
review text: 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 \(\alpha\in (0,1)\).
reviewed by: A. M. Plichko
Mathematics Subject Classification ID: 46C15
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 51B20
zbMATH DE Number: 5651843
zbMATH Keywords: arc length
zbMATH Keywords: chord length
zbMATH Keywords: Euclidean plane
zbMATH Keywords: inner product space
zbMATH Keywords: Minkowski plane
zbMATH Keywords: normed linear space
MaRDI profile type: Publication
full work available at URL: s10998-008-8207-6
OpenAlex ID: W2084527903
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DOI: 10.1007/S10998-008-8207-6
description: scientific article; zbMATH DE number 5651843