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Covering Discs in Minkowski Planes

Published online by Cambridge University Press:  20 November 2018

Horst Martini  and
Margarita Spirova
Horst Martini
Affiliation:
Fakultät für Mathematik, TU Chemnitz, D-09107 Chemnitz, Germany e-mail: horst.martini@mathematik.tu-chemnitz.de
Margarita Spirova
Affiliation:
Faculty of Mathematics and Informatics, University of Sofia, 5 James Bourchier, 1164 Sofia, Bulgaria e-mail: spirova@fmi.uni-sofia.bg
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Abstract

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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 by $k$ unit circles. In particular, we study the cases $k\,=\,3$, $k\,=\,4$, and $k\,=\,7$. For $k\,=\,3$ and $k\,=\,4$, 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.

Keywords

46B2052A2152C15affine regular polygonbisectorcircle covering problemcircumradiusd-segmentMinkowski plane(strictly convex) normed plane
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 3 , 01 September 2009 , pp. 424 - 434
Copyright
Copyright © Canadian Mathematical Society 2009

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