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Covering Discs in Minkowski Planes |
Published:2009-09-01
Printed: Sep 2009
Printed: Sep 2009
- Horst Martini
- Margarita Spirova
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 |
| MSC Classifications: |
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] |