On Minimal Tilings with Convex Cells Each Containing a Unit Ball

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


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 \(13.8564\ldots\).

Key words

Tiling Convex cell Unit sphere packing Average edge curvature Foam problem 

Subject Classifications

05B40 05B45 52B60 52C17 52C22 


  1. 1.
    Ambrus, G., Fodor, F.: A new lower bound on the surface area of a Voronoi polyhedron. Period. Math. Hungar. 53(1–2), 45–58 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Besicovitch, A.S., Eggleston, H.G.: The total length of the edges of a polyhedron. Q. J. Math. Oxford Ser. 2(8), 172–190 (1957)MathSciNetCrossRefGoogle Scholar
  3. 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, 131–143 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bezdek, K.: On rhombic dodecahedra. Contrib. Alg. Geom. 41(2), 411–416 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bezdek, K.: A lower bound for the mean width of Voronoi polyhedra of unit ball packings in E 3. Arch. Math. 74(5), 392–400 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bezdek, K., Daróczy-Kiss, E.: Finding the best face on a Voronoi polyhedron – the strong dodecahedral conjecture revisited. Monatsh. Math. 145(3), 191–206 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dekster, B.V.: An extension of Jung’s theorem. Isr. J. Math. 50(3), 169–180 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Fejes Tóth, L.: Regular Figures. Pergamon Press, New York (1964)zbMATHGoogle Scholar
  9. 9.
    Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. 162(3), 1065–1185 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hales, T.C., Ferguson, S.P.: A formulation of the Kepler conjecture. Discrete Comput. Geom. 36(1), 21–69 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hales, T.C.: Sphere packings III, extremal cases. Discrete Comput. Geom. 36(1), 71–110 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hales, T.C.: Sphere packings IV, detailed bounds. Discrete Comput. Geom. 36(1), 111–166 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hales, T.C.: Sphere packings VI, Tame graphs and linear programs. Discrete Comput. Geom. 36(1), 205–265 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hales, T.C.: Dense Sphere Packings. A Blueprint for Formal Proofs. Cambridge University Press, Cambridge (2012)zbMATHCrossRefGoogle Scholar
  15. 15.
    Kertész, G.: On totally separable packings of equal balls. Acta Math. Hungar. 51(3–4), 363–364 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Morgan, F.: Geometric Measure Theory - A Beginner’s Guide. Elsevier – Academic Press, Amsterdam (2009)zbMATHGoogle Scholar
  17. 17.
    Rogers, C.A.: Packing and Covering. Cambridge University Press, Cambridge (1964)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  2. 2.Department of MathematicsUniversity of PannoniaHungaryCanada

Personalised recommendations