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A CHARACTERISATION OF EUCLIDEAN NORMED PLANES VIA BISECTORS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 20 August 2018, pp. 121-129
- Print publication:
- February 2019
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- Article
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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.
