A CHARACTERISATION OF EUCLIDEAN NORMED PLANES VIA BISECTORS. Issue 1 (20th August 2018)

• Record Type: Journal Article
• Authors: CABELLO SÁNCHEZ, JAVIER; GORDILLO-MERINO, ADRIÁN
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 99:Issue 1(2019)
• Page Start: 121
• Page End: 129
• Publication Date: 2018-08-20
• Subjects: primary 46B20, -- secondary 52A10, -- 52A21; isosceles orthogonality, -- strictly convex normed spaces, -- Euclidean planes, -- bisectors; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972718000758
• Languages: English
• ISSNs: 0004-9727

Abstract

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.