Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 44, No. 2, pp. 493498 (2003) 

On $k^+$neighbour packings and onesided Hadwiger configurationsKároly Bezdek and Peter BrassDepartment of Geometry, Eötvös University, H1117 Budapest, Pázmány Péter sétány 1/c, email: kbezdek@ludens.elte.hu Department of Computer Science, The City College, CUNY, 138th Street at Convent Avenue, New York NY10031, USA, email: peter@cs.ccny.cuny.eduAbstract: We show that in $d$dimensional Euclidean space the maximum number of nonoverlapping translates of a $d$dimensional convex body $K$ that can touch $K$ and can lie in a closed supporting halfspace of $K$ is always at most $2\cdot 3^{d1} 1$, with this bound to be reached only if $K$ is an affine image of a $d$cube. Such onesided 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 $k\geq 2\cdot 3^{d1}$; and there is a $(2\cdot3^{d1}1)^+$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). Full text of the article:
Electronic version published on: 1 Aug 2003. This page was last modified: 4 May 2006.
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