A characterisation of Euclidean normed planes via bisectors

Our main result states that whenever we have a non-Euclidean norm \(\|\cdot\|\) on a two-dimensional vector space \(X\), there exists some \(x\neq 0\) such that for every \(\lambda\neq 1, \lambda>0\), there exist \(y, z\in X\) verifying that \(\|y\|=\lambda\|x\|\), \(z\neq 0\), and \(z\) belongs...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Authors: Javier Cabello Sánchez, Gordillo-Merino, Adrián
Format: Article
Language:English
Subjects:
Planes
Vector spaces
Online Access:Get full text
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Summary:Our main result states that whenever we have a non-Euclidean norm \(\|\cdot\|\) on a two-dimensional vector space \(X\), there exists some \(x\neq 0\) such that for every \(\lambda\neq 1, \lambda>0\), there exist \(y, z\in X\) verifying that \(\|y\|=\lambda\|x\|\), \(z\neq 0\), and \(z\) belongs to the bisectors \(B(-x,x)\) and \(B(-y,y)\). Throughout this paper we also state and prove some other simple but maybe useful results about the geometry of the unit sphere of strictly convex planes.
ISSN:2331-8422
DOI:10.48550/arxiv.2402.05769