Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-19T05:32:23.687Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A STRONG VERSION OF THE KEPLER CONJECTURE

Part of: Discrete geometry Polytopes and polyhedra Designs and configurations

Published online by Cambridge University Press:  01 August 2012

Károly Bezdek
Károly Bezdek*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW Calgary, Canada T2N 1N4 Department of Mathematics, University of Pannonia, Veszprém, Hungary Institute of Mathematics, Eötvös University, Budapest, Hungary (email: [email protected])

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We raise and investigate the following problem which one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least

MSC classification

Secondary: 05B40: Packing and covering 05B45: Tessellation and tiling problems 52B60: Isoperimetric problems for polytopes 52C17: Packing and covering in $n$ dimensions 52C22: Tilings in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 23 - 30
Copyright
Copyright © University College London 2012

References

[1]Ambrus, G. and Fodor, F., A new lower bound on the surface area of a Voronoi polyhedron. Period. Math. Hungar. 53/1–2 (2006), 45–58.CrossRefGoogle Scholar
[2]Besicovitch, A. S. and Eggleston, H. G., The total length of the edges of a polyhedron. Q. J. Math. 2/8 (1957), 172–190.CrossRefGoogle Scholar
[3]Bezdek, K., On a stronger form of Rogers’s lemma and the minimum surface area of Voronoi cells in unit ball packings. J. reine angew. Math. 518 (2000), 131–143.Google Scholar
[4]Bezdek, K., On rhombic dodecahedra. Contrib. Algebra Geom. 41/2 (2000), 411–416.Google Scholar
[5]Bezdek, K. and Daróczy-Kiss, E., Finding the best face on a Voronoi polyhedron—the strong dodecahedral conjecture revisited. Monatsh. Math. 145/3 (2005), 191–206.CrossRefGoogle Scholar
[6]Dekster, B. V., An extension of Jung’s theorem. Israel J. Math. 50/3 (1985), 169–180.CrossRefGoogle Scholar
[7]Fejes Tóth, L., Regular Figures, Pergamon Press (New York, 1964).Google Scholar
[8]Hales, T. C., The strong dodecahedral conjecture and Fejes Tóth’s contact conjecture, arXiv:1110.0402v1 [math.MG] (2011), 1–11.Google Scholar
[9]Hales, T. C., A proof of the Kepler conjecture. Ann. of Math. (2) 162/3 (2005), 1065–1185.CrossRefGoogle Scholar
[10]Hales, T. C., Sphere packings III, Extremal cases. Discrete Comput. Geom. 36/1 (2006), 71–110.CrossRefGoogle Scholar
[11]Hales, T. C., Sphere packings IV, Detailed bounds. Discrete Comput. Geom. 36/1 (2006), 111–166.CrossRefGoogle Scholar
[12]Hales, T. C., Sphere packings VI, Tame graphs and linear programs. Discrete Comput. Geom. 36/1 (2006), 205–265.CrossRefGoogle Scholar
[13]Hales, T. C. and Ferguson, S. P., A formulation of the Kepler conjecture. Discrete Comput. Geom. 36/1 (2006), 21–69.CrossRefGoogle Scholar
[14]Kertész, G., On totally separable packings of equal balls. Acta Math. Hungar. 51/3–4 (1988), 363–364.CrossRefGoogle Scholar
[15]Morgan, F., Geometric Measure Theory – A Beginner’s Guide, 4th edn, Elsevier/Academic Press (Amsterdam, 2009).Google Scholar
[16]Rogers, C. A., Packing and Covering, Cambridge University Press (Cambridge, 1964).Google Scholar