Improving Rogers' upper bound for the density of unit ball packings via estimating the surface area of Voronoi cells from below in Euclidean d-space for all d ≥ 8

Discrete and Computational Geometry

Volumn 28, Issue 1, 2002, Pages 75-106

  Bezdek, Károly ^a,b  

^a EÖTVÖS LORÁND UNIVERSITY   (Hungary)

^b Department of Obstetrics Gynecology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036665954     PISSN: 01795376     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00454-001-0095-y     Document Type: Article

References (26)

1. K. M. Ball, A lower bound for the optimal density of lattice packings, Duke Math. J. 68 (1992), 217-221.
2. E. Baranovskii, On packing n-dimensional Euclidean space by equal spheres, Izv. Vyssh. Uchebn. Zaved. Mat. 39(2) (1964), 14-24.
3. A. Bezdek, K. Bezdek and R. Connelly, Finite and uniform stability of sphere packings, Discrete Comput. Geom. 20 (1998), 111-130.
4. K. Bezdek, 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.
5. K. Bezdek and R. Connelly, Two-distance preserving functions from Euclidean space, Period. Math. Hungar. 39(1-3) (1999), 185-200.
6. K. Böröczky, Packing equal spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32(3-4) (1978), 243-261.
7. H. Cohn and N. Elkies, New bounds on sphere packings, I, published electronically at http://arXiv.org/abs/math.MG/011009 (September 30, 2001), pp. 1-21.
8. J. H. Conway and N. J. A. Sloane, Laminated lattices, Ann. of Math. 116 (1982), 593-620.
9. J. H. Conway and N. J. A. Sloane, The antipode construction for sphere packings, Invent. Math. 123 (1996), 309-313.
10. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd edn., Springer-Verlag, New York, 1998.
11. M. H. Dauenhauer and H. J. Zassenhaus, Local optimality of critical lattice sphere-packing of regular tetrahedra, Discrete Math. 64 (1987), 129-146.
12. G. Fejes Tóth and W. Kuperberg, Packing and covering with convex sets, in Handbook of Convex Geometry (P. M. Gruber and J. M. Wills, eds.), Elsevier, Amsterdam, 1993, pp. 799-860.
13. T. C. Hales, Remarks on the density of sphere packings in three dimensions, Combinatorica 13 (1993), 181-197.
14. T. C. Hales, Sphere packings, I, Discrete Comput. Geom. 17 (1997), 1-51.
15. T. C. Hales, Sphere packings, II, Discrete Comput. Geom. 18 (1997), 135-149.
16. T. C. Hales, The Kepler conjecture, manuscript, 1998.
17. W.-Y. Hsiang, The optimal density of sphere packings in dimension 8 and the uniqueness theorem on final packings with optimal density, Berkeley Mathematics Department Colloquium (March 15, 2001).
18. G. A. Kabatjanskii and V. I. Levenštein, Bounds for packings on a sphere and in space (in Russian), Problemy Peredachi Informatsii 14 (1978), 3-25; English translation in Problems Informat. Transmission 14 (1978),1-17.
19. F. R. Kschischang and S. Pasupathy, Some ternary and quaternary codes and associated sphere packings, IEEE Trans. Inform. Theory 38 (1992), 227-246.
20. J. Leech and N. J. A. Sloane, Sphere packing and error-correcting codes, Canad. J. Math. 23 (1971), 718-745.
21. D. J. Muder, A new bound on the local density of sphere packings, Discrete Comput. Geom. 10 (1993), 351-375.
22. G. Nebe and N. J. A. Sloane, Table of densest packings presently known, published electronically at http://www.research.att.com/njas/lattices/density.html.
23. C. A. Rogers, The packing of equal spheres, Proc. London Math. Soc. 3(8) (1958), 609-620.
24. C. A. Rogers, Packing and Covering, Cambridge University Press, Cambridge, 1964.
25. N. J. A. Sloane, The sphere packing problem, Doc. Math. J. DMV III (1998), 387-396.
26. A. Vardy, A new sphere packing in 20 dimensions, Invent. Math. 121 (1995), 119-133.