Covering Discs in Minkowski Planes

Canadian mathematical bulletin

关键词: affine regular polygon;    bisector;    circle covering problem;    circumradius;    $d$-segment;    Minkowski plane;    (strictly convex) normed plane;   

DOI  :  10.4153/CMB-2009-046-2

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We investigate the following version of the circle coveringproblem in strictly convex (normed or) Minkowski planes: to covera circle of largest possible diameter by $k$ unit circles. Inparticular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$and $k=4$, the diameters under consideration are described interms of side-lengths and circumradii of certain inscribed regulartriangles or quadrangles. This yields also simple explanations ofgeometric meanings that the corresponding homothety ratios have.It turns out that basic notions from Minkowski geometry play anessential role in our proofs, namely Minkowskian bisectors,$d$-segments, and the monotonicity lemma.

【 授权许可】

Unknown