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1. CMB 2009 (vol 52 pp. 424)
Covering Discs in Minkowski Planes 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 Categories:46B20, 52A21, 52C15 |
2. CMB 2009 (vol 52 pp. 451)
Indecomposable Coverings We prove that for every $k>1$, there exist $k$-fold coverings of the
plane (i) with strips, (ii) with axis-parallel rectangles, and
(iii) with homothets of any fixed concave quadrilateral, that cannot
be decomposed into two coverings. We also construct for every
$k>1$ a set of points $P$ and a family of disks $\cal D$ in the
plane, each containing at least $k$ elements of $P$, such that, no
matter how we color the points of $P$ with two colors,
there
exists a disk $D\in{\cal D}$ all of whose points are of the same
color.
Categories:52C15, 05C15 |
3. CMB 1999 (vol 42 pp. 380)
Asymptotic Behavior of Optimal Circle Packings in a Square A lower bound on the number of points that can be placed in a
square of side $\sigma$ such that no two points are within unit
distance from each other is proven. The result is constructive,
and the series of packings obtained contains many conjecturally
optimal packings.
Keywords:asymptotic bound, circle packing Category:52C15 |