Covering Discs in Minkowski Planes
Horst Martini, Margarita Spirova · Canadian Mathematical Bulletin · 2009
Abstract We investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter bykunit circles. In particular, we study the casesk= 3,k= 4, andk= 7. Fork= 3 andk= 4, the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors,d-segments, and the monotonicity lemma.