On orthogonal chords in normed planes
type: Item
instance of: scholarly article
title: On orthogonal chords in normed planes
zbMATH Open document ID: 1214.52001
DOI: 10.1216/RMJ-2011-41-1-23
author: Javier Alonso
author: Horst Martini
author: Zokhrab Mustafaev
published in: Rocky Mountain Journal of Mathematics
publication date: 2011-03-01T00:00:00Z
review text: 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.
reviewed by: Rita Pini
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5858877
zbMATH Keywords: Birkhoff orthogonality
zbMATH Keywords: convex figure
zbMATH Keywords: constant width
zbMATH Keywords: isosceles orthogonality
zbMATH Keywords: maximal chord
zbMATH Keywords: normed plane
MaRDI profile type: Publication
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full work available at URL: rmj-2011-41-1-23
OpenAlex ID: W2059518611
description: scientific article; zbMATH DE number 5858877