An example concerning the translative kissing number of a convex body
type: Item
instance of: scholarly article
title: An example concerning the translative kissing number of a convex body
zbMATH Open document ID: 0813.52015
DOI: 10.1007/BF02574374
published in: Discrete \& Computational Geometry
publication date: 1994-09-26T00:00:00Z
full work available at URL: https://eudml.org/doc/131325
review text: The translative kissing number \(N_ T(K)\) of the convex body \(K\) is the largest \(N\) such that \(N\) translates of \(K\) with disjoint interiors can all touch \(K\). The lattice kissing number \(N_ L(K)\) is the largest such that \(N\) translates of \(K\) in a lattice packing can all touch \(K\). In 1961, Grünbaum showed that \(N_ T(K)= N_ L(K)\) for all \(K\) in \(E^ 2\); in 1971 Watson found a counterexample for the sphere in \(E^ 9\). The author presents a nice counterexample for a convex polyhedron in \(E^ 3\). The unit cube touches 26 translates in the face-to-face lattice packing, eight of them by a tenuous vertex-to-vertex contact. By slightly faceting four vertices (the endpoints of two opposite edges), the author creates a polyhedron that cannot have more than 22 neighbours in a lattice packing. However, a non-lattice configuration is given in which the polyhedron touches 24 translates.
Mathematics Subject Classification ID: 52C07
Mathematics Subject Classification ID: 90C05
zbMATH DE Number: 644705
zbMATH Keywords: polytope
zbMATH Keywords: translative kissing number
zbMATH Keywords: convex body
zbMATH Keywords: lattice kissing number
author: Chuan Ming Zong
reviewed by: Robert Dawson
MaRDI profile type: Publication
cites work: Q4039784
cites work: Abschätzungen für die Anzahl der konvexen Körper, die einen konvexen Körper berühren
cites work: On a conjecture of H. Hadwiger
cites work: Q5618419
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description: scientific article; zbMATH DE number 644705