A CHARACTERISATION OF EUCLIDEAN NORMED PLANES VIA BISECTORS
Javier Cabello Sánchez, Adrián Gordillo-Merino · Bulletin of the Australian Mathematical Society · 2018
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 $x eq 0$ such that for every $\unicode[STIX]{x1D706} eq 1$ , $\unicode[STIX]{x1D706}>0$ , there exist $y,z\in X$ satisfying $\Vert y\Vert =\unicode[STIX]{x1D706}\Vert x\Vert$ , $z eq 0$ 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.