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Covering Discs in Minkowski Planes

  Published:2009-09-01
 Printed: Sep 2009
  • Horst Martini
  • Margarita Spirova
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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 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: affine regular polygon, bisector, circle covering problem, circumradius, $d$-segment, Minkowski plane, (strictly convex) normed plane affine regular polygon, bisector, circle covering problem, circumradius, $d$-segment, Minkowski plane, (strictly convex) normed plane
MSC Classifications: 46B20, 52A21, 52C15 show english descriptions Geometry and structure of normed linear spaces
Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
Packing and covering in $2$ dimensions [See also 05B40, 11H31]
46B20 - Geometry and structure of normed linear spaces
52A21 - Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
52C15 - Packing and covering in $2$ dimensions [See also 05B40, 11H31]
 

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