Discrete & Computational Geometry

, Volume 8, Issue 2, pp 109–130 | Cite as

On the covering multiplicity of lattices

  • J. H. Conway
  • N. J. A. Sloane


Let the lattice Λ have covering radiusR, so that closed balls of radiusR around the lattice points just cover the space. The covering multiplicityCM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity for ann-dimensional lattice isn ifn≤8, and conjecture that it exceedsn in all other cases. We determine the covering multiplicity of the Leech lattice and of the latticesIn, An, Dn, En and their duals for small values ofn. Although it appears thatCM(In)=2n−1 ifn≤33, asn → ∞ we haveCM(In)∼2.089... n . The results have application to numerical integration.


Open Ball Closed Ball Deep Hole Theta Series Dual Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    N. Bourbaki,Groupes et Algèbres de Lie, Chapitres 4, 5 et 6. Hermann, Paris, 1968.Google Scholar
  2. 2.
    J. H. Conway, R. A. Parker, and N. J. A. Sloane, The covering radius of the Leech lattice,Proc. Roy. Soc. London Ser. A,380 (1982), 261–290. A revised version appears as Chapter 23 of [3].MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    J. H. Conway and N. J. A. Sloane,Sphere Packings, Lattices and Groups, Springer-Verlag, New York, 1988.MATHCrossRefGoogle Scholar
  4. 4.
    J. H. Conway and N. J. A. Sloane, The cell structures of lattices, inMiscellanea mathematica, ed. P. Hiltonet al., Springer-Verlag, New York, 1991, pp. 71–107.CrossRefGoogle Scholar
  5. 5.
    H. S. M. Coxeter,Regular Polytopes, 3rd edn., Dover, New York, 1973.Google Scholar
  6. 6.
    D. J. Elzinga and D. W. Hearn, Geometrical solutions for some minimax location problems,Transportation Sci.,6 (1972), 379–394.MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. J. Elzinga and D. W. Hearn, The minimum covering sphere problem,Management Sci.,19(1) (1972), 96–104.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    D. J. Elzinga and D. W. Hearn, The minimum sphere covering a convex polyhedron,Naval Res. Logist. Quart.,21 (1974), 715–718.MathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A modeling language for mathematical programming,Management Sci.,36 (1990), 519–554.MATHCrossRefGoogle Scholar
  10. 10.
    J. E. Humphreys,Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.MATHCrossRefGoogle Scholar
  11. 11.
    Mathlab Group, MACSYMA Reference Manual, Version 10, Laboratory for Computer Science, MIT, Cambridge, MA, 1983.Google Scholar
  12. 12.
    J. E. Mazo and A. M. Odlyzko, Lattice points in high-dimensional spheres,Monatsh. Math.,110 (1990), 47–61.MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    N. Megiddo and K. J. Supowit, On the complexity of some common geometric location problems,SIAM J. Comput.,13 (1984), 182–196.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    B. A. Murtagh and M. A. Saunders, Large-scale linearly constrained optimization,Math. Programming,14 (1978), 41–72.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    B. A. Murtagh and M. A. Saunders, MINOS 5.1 User's Guide, Technical Report SOL 83-20R, Dept. Operations Research, Stanford University, Stanford, CA, Jan. 1987.Google Scholar
  16. 16.
    F. P. Preparata and M. I. Shamos,Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.CrossRefGoogle Scholar
  17. 17.
    W. D. Smith, Studies in Computational Geometry Motivated by Mesh Generation, Ph.D. Dissertation, Program in Applied and Computational Mathematics, Princeton University, Sept. 1988.Google Scholar
  18. 18.
    J. M. Sullivan, A Crystalline Approximation Theorem for Hypersurfaces, Ph.D. Dissertation, Department of Mathematics, Princeton University, 1990.Google Scholar
  19. 19.
    R. T. Worley, The Voronoi region ofE *6,J. Austral. Math. Soc. Ser. A 48 (1987), 268–278.MathSciNetCrossRefGoogle Scholar
  20. 20.
    R. T. Worley, The Voronoi region ofE *7,SIAM J. Discrete Math.,1 (1988), 134–141.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • J. H. Conway
    • 1
  • N. J. A. Sloane
    • 2
  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Mathematical Sciences Research CenterAT&T Bell LaboratoriesMurray HillUSA

Personalised recommendations