On the Hadwiger numbers of starlike disks

Details On the Hadwiger numbers of starlike disks On the Hadwiger numbers of starlike disks

2010-02-20

arxiv.org/abs/1002.3918v3

Mathematics

Metric Geometry

13 pages, 8 figures

Scientific paper

10.1016/j.ejc.2011.05.004

The Hadwiger number $H(J)$ of a topological disk $J$ in $\Re^2$ is the maximal number of pairwise nonoverlapping translates of $J$ that touch $J$. It is well known that for a convex disk, this number is six or eight. A conjecture of A. Bezdek., K. and W. Kuperberg says that the Hadwiger number of a starlike disk is at most eight. A. Bezdek proved that this number is at most seventy five for any starlike disk. In this note, we prove that the Hadwiger number of a starlike disk is at most thirty five. Furthermore, we show that the Hadwiger number of a topological disk $J$ such that $(\conv J) \setminus J$ is connected, is six or eight.

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