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A CHARACTERISATION OF EUCLIDEAN NORMED PLANES VIA BISECTORS

Part of: Normed linear spaces and Banach spaces; Banach lattices General convexity

Published online by Cambridge University Press:  20 August 2018

JAVIER CABELLO SÁNCHEZ  and
ADRIÁN GORDILLO-MERINO
JAVIER CABELLO SÁNCHEZ*
Affiliation:
Departamento de Matemáticas, Universidad de Extremadura, Avda. de Elvas s/n, 06006 Badajoz, Spain email coco@unex.es
ADRIÁN GORDILLO-MERINO
Affiliation:
Departamento de Matemáticas, Universidad de Extremadura, Avda. de Elvas s/n, 06006 Badajoz, Spain email adgormer@unex.es
*
✉coco@unex.es
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Abstract

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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.

Keywords

isosceles orthogonalitystrictly convex normed spacesEuclidean planesbisectors

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 52A10: Convex sets in $2$ dimensions (including convex curves) 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
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.

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