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A CHARACTERISATION OF EUCLIDEAN NORMED PLANES VIA BISECTORS
Published online by Cambridge University Press: 20 August 2018
Abstract
Our main result states that whenever we have a non-Euclidean norm $\Vert \cdot \Vert$ on a two-dimensional vector space
$X$, there exists some
$x\neq 0$ such that for every
$\unicode[STIX]{x1D706}\neq 1$,
$\unicode[STIX]{x1D706}>0$, there exist
$y,z\in X$ satisfying
$\Vert y\Vert =\unicode[STIX]{x1D706}\Vert x\Vert$,
$z\neq 0$ and
$z$ belongs to the bisectors
$B(-x,x)$ and
$B(-y,y)$. We also give several results about the geometry of the unit sphere of strictly convex planes.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 121 - 129
- Copyright
- © 2018 Australian Mathematical Publishing Association Inc.
Footnotes
The first author was supported in part by DGICYT project MTM2016⋅76958⋅C2⋅1⋅P (Spain) and Junta de Extremadura programs GR⋅15152 and IB⋅16056; the second author was partially supported by Junta de Extremadura and FEDER funds.
References
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