Circumradius versus side lengths of triangles in linear normed spaces

Gennadiy Averkov¹

¹ Institute of Algebra and Geometry Faculty of Mathematics Otto-von-Guericke University of Magdeburg Universitätsplatz 2 D-39106 Magdeburg, Germany

Colloquium Mathematicum, Tome 107 (2007) no. 2, pp. 273-285

Résumé

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.

Détail

DOI : 10.4064/cm107-2-7

Keywords: given planar convex body centered origin denote cal minkowski plane two dimensional linear normed space unit ball triangle cal denote least possible radius minkowskian ball enclosing remark terminology location science optimum minimax location problem distance induced vertices existing facilities see instance references therein using methods linear algebra convex geometry lower upper bound arbitrary planar convex body centered origin subseteq cal arbitrary triangle given minkowskian side lengths obtain further results geometry triangles minkowski planes which either corollaries main result statements needed proof main result