A characterisation of Euclidean normed planes via bisectors
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909099104403456 |
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| author | Sánchez, Javier Cabello Gordillo-Merino, Adrián |
| author_facet | Sánchez, Javier Cabello Gordillo-Merino, Adrián |
| contents | 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 $λ\neq 1, λ>0$, there exist $y, z\in X$ verifying that $\|y\|=λ\|x\|$, $z\neq 0$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05769 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterisation of Euclidean normed planes via bisectors Sánchez, Javier Cabello Gordillo-Merino, Adrián Metric Geometry 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 $λ\neq 1, λ>0$, there exist $y, z\in X$ verifying that $\|y\|=λ\|x\|$, $z\neq 0$, 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. |
| title | A characterisation of Euclidean normed planes via bisectors |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2402.05769 |