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On Minimal Tilings with Convex Cells Each Containing a Unit Ball

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Discrete Geometry and Optimization

Part of the book series: Fields Institute Communications ((FIC,volume 69))

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Abstract

We investigate the following problem that 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 edge curvature of the cells? In particular, we prove that the average edge curvature in question is always at least \(13.8564\ldots\).

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada

    Károly Bezdek

  2. Department of Mathematics, University of Pannonia, Veszprém, Hungary, Canada

    Károly Bezdek

Authors
  1. Károly Bezdek
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Correspondence to Károly Bezdek .

Editor information

Editors and Affiliations

  1. , Mathematics & Statistics, University of Calgary, University Drive NW, Calgary, T2N 1N4, Alberta, Canada

    Karoly Bezdek

  2. , Department of Computing, McMaster University, Main Street West 1280, Hamilton, L8S 4K1, Ontario, Canada

    Antoine Deza

  3. School of Engineering, Department of Management Science and Eng, Stanford University, Via Ortega 475, Stanford, 94305, California, USA

    Yinyu Ye

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Bezdek, K. (2013). On Minimal Tilings with Convex Cells Each Containing a Unit Ball. In: Bezdek, K., Deza, A., Ye, Y. (eds) Discrete Geometry and Optimization. Fields Institute Communications, vol 69. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00200-2_4

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