Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 50, No. 1, pp. 249-257 (2009)

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On the Hadwiger numbers of centrally symmetric starlike disks

Zsolt Lángi

Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4, e-mail:

Abstract: The Hadwiger number $H(S)$ of a topological disk $S$ in ${\mathbb R}^2$ is the maximal number of pairwise nonoverlapping translates of $S$ that touch $S$. A conjecture of A. Bezdek., K. and W. Kuperberg [BKK] states that this number is at most eight for any starlike disk. A. Bezdek [B] proved that the Hadwiger number of a starlike disk is at most seventy five. In this note, we prove that the Hadwiger number of any centrally symmetric starlike disk is at most twelve.

[BKK] Bezdek, A.; Kuperberg, K.; Kuperberg, W.: Mutually contiguous translates of a plane disk. Duke Math. J. {\bf 78} (1995), 19--31. [B] Bezdek, A.: On the Hadwiger number of a starlike disk. In: Bárány, I. (ed.) et al., Intuitive Geometry Bolyai Soc. Math. Studies {\bf 6} (1997), 237--245.

Keywords: topological disk, starlike disk, touching, Hadwiger number

Classification (MSC2000): 52A30, 52A10, 52C15

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Electronic version published on: 29 Dec 2008. This page was last modified: 28 Jan 2013.

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