Circumradius versus Side Lengths of Triangles in Linear Normed Spaces

Given a planar convex body B centered at the origin, we denote by M(B) the Minkowski plane (i.e., two-dimensional linear normed space) with the unit ball B. For a triangle T in M(B) we denote by RB(T ) the least possible radius of a Minkowskian ball enclosing T. We remark that in terms of location science RB(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 from linear algebra and convex geometry we find the lower and the upper bound of RB(T ) for the case when B is an arbitrary planar convex bodies centered at the origin and T ⊆ M(B) is an arbitrary triangle with given Minkowskian side lengths a1, a2, a3. Additionally, 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. Mathematics Subject Classification (2000): 52A21, 52B12, 52C15