Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 44, No. 2, pp. 493-498 (2003)

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On $k^+$-neighbour packings and one-sided Hadwiger configurations

Károly Bezdek and Peter Brass

Department of Geometry, Eötvös University, H-1117 Budapest, Pázmány Péter sétány 1/c, e-mail: Department of Computer Science, The City College, CUNY, 138th Street at Convent Avenue, New York NY-10031, USA, e-mail:

Abstract: We show that in $d$-dimensional Euclidean space the maximum number of non-overlapping translates of a $d$-dimensional convex body $K$ that can touch $K$ and can lie in a closed supporting half-space of $K$ is always at most $2\cdot 3^{d-1} -1$, with this bound to be reached only if $K$ is an affine image of a $d$-cube. Such one-sided Hadwiger configurations occur at the boundary of finite packings or near the holes in packings of density 0, so this implies that in $d$-dimensional Euclidean space any $k^{+}$-neighbour packing by translates of a $d$-dimensional convex body has positive density for all $k\geq 2\cdot 3^{d-1}$; and there is a $(2\cdot3^{d-1}-1)^+$-neighbour packing by translates of a $d$-cube that has density 0. (A packing is called a $k^+$-neighbour packing if each packing element has at least $k$ neighbours.) This answers an old question of L. Fejes Tóth (1973).

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Electronic version published on: 1 Aug 2003. This page was last modified: 4 May 2006.

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