A characterisation of Euclidean normed planes via bisectors

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 $x\ne 0$ such that for every $\lambda \ne 1,\lambda >0$, there exist $y,z\in X$ verifying that $\Vert y\Vert =\lambda \Vert x\Vert$, $z\ne 0$, 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.