## Circumradius versus side lengths of triangles in linear normed spaces

### Volume 107 / 2007

#### Abstract

Given a planar convex body $B$ centered at the origin, we denote by ${\cal M}^2(B)$ the Minkowski plane (i.e., two-dimensional linear normed space) with the unit ball $B.$ For a triangle $T$ in ${\cal M}^2(B)$ we denote by $R_B(T)$ the least possible radius of a Minkowskian ball enclosing $T.$ We remark that in the terminology of location science $R_B(T)$ is the optimum of the minimax location problem with distance induced by $B$ and vertices of $T$ as existing facilities (see, for instance, [HM03] and the references therein). Using methods of linear algebra and convex geometry we find the lower and upper bound of $R_B(T)$ for the case when $B$ is an arbitrary planar convex body centered at the origin and $T \subseteq {\cal M}^2(B)$ is an arbitrary triangle with given Minkowskian side lengths $a_1, a_2, a_3.$ We also obtain some further results from the geometry of triangles in Minkowski planes, which are either corollaries of the main result or statements needed in the proof of the main result.