Inventiones mathematicae

, Volume 105, Issue 1, pp 613–639 | Cite as

On the packing densities of superballs and other bodies

  • N. D. Elkies
  • A. M. Odlyzko
  • J. A. Rush


A method of obtaining improvements to the Minkowski-Hlawka bound on the lattice-packing density for many convex bodies symmetrical through the coordinate hyperplanes, described by Rush [18], is generalized so that centrally symmetric convex bodies can be treated as well. The lower bounds which arise are very good.

The technique is applied to various shapes, including the classicallσ-ball,
$$\left\{ {x \in R^n :|x_1 |^\sigma + |x_2 |^\sigma + ... + |x_n |^\sigma \leqq 1} \right\},$$
for σ≧1. This generalizes the earlier work of Rush and Sloane [17] in which σ was required to be an integer. The superball above can be lattice packed to a density of(b/2)n+0(1) for largen, where
$$b = \mathop {\sup }\limits_{t > 0} \frac{{\int\limits_{x = - \infty }^\infty {e^{ - | tx |^\sigma d x} } }}{{\sum\limits_{k = - \infty }^\infty {e^{ - | tx |^\sigma } } }}.$$
This is as good as the Minkowski-Hlawka bound for 1≦σ≦2, and better for σ>2.
An analogous density bound is established for superballs of the shape
$$\left\{ {x \in R^n :f(x_1 ,...,x_k )^\sigma + f(x_{k + 1} ,...,x_{2k} )^\sigma + ... + f(x_{n - k + 1} ,...,x_n )^\sigma \leqq 1} \right\}, k|n,$$
wheref is the Minkowski distance function associated with a bounded, convex, centrally symmetric,k-dimensional body.

Finally, we consider generalized superballs for which the defining inequality need not even be homogeneous. For these bodies as well, it is often possible to improve on the Minkowski-Hlawka bound.


Lower Bound Distance Function Packing Density Convex Body Symmetric Convex 
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  1. 1.
    de Bruijn, N.G.: Asymptotic Methods in Analysis. New York: Dover 1981 (Reprint)Google Scholar
  2. 2.
    Cassels, J.W.S.: An Introduction to the Geometry of Numbers. Berlin Heidelberg New York: Springer 1959Google Scholar
  3. 3.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. Berlin Heidelberg New York: Springer 1987Google Scholar
  4. 4.
    Erdös, P., Gruber, P.M., Hammer, J.: Lattice Points. Pitman Monograph 39, Longman Scientific. New York: John Wiley & Sons, 1989Google Scholar
  5. 5.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. New York: Academic Press 1980 (Translated from Russian)Google Scholar
  6. 6.
    Gruber, P.M., Lekkerkerker, C.G.: Geometry of Numbers. Amsterdam: North-Holland Elsevier 1987 (This is an updated version of [10])Google Scholar
  7. 7.
    Hlawka, E.: Zur Geometrie der Zahlen. Math. Z.49, 285–312 (1943)Google Scholar
  8. 8.
    Kabatiansky, G.A., Levenshtein, V.I.: Bounds for packings on a sphere and in space (in Russian). Probl. Peredachy Inf.14, 3–25 (1978); (English translation in) Probl. Inf. Transm.14, 1–17 (1978)Google Scholar
  9. 9.
    Leech, J., Sloane, N.J.A.: Sphere packing and error-correcting codes. Can. J. Math.23, 718–745 (1971)Google Scholar
  10. 10.
    Lekkerkerker, C.G.: Geometry of Numbers, Groningen: Wolters-Noordhoff 1969Google Scholar
  11. 11.
    Litsin, S.N., Tsfasman, M.A.: Algebraic-geometric and number-theoretic packings of spheres (in Russian). Usp. Mat. Nauk40, 185–186 (1985)Google Scholar
  12. 12.
    Mazo, J.E., Odlyzko, A.M.: Lattice points in high-dimensional spheres. Monatsh. Math.110, 47–61 (1990)Google Scholar
  13. 13.
    Minkowski, H.: Geometrie der Zahlen, I, Leipzig: Teubner 1896Google Scholar
  14. 14.
    Minkowski, H.: eesammelte Abhandlungen. New York: Chelsea 1969 (Reprint)Google Scholar
  15. 15.
    Rogers C.A.: Packing and Covering. Cambridge: Cambridge University Press 1964Google Scholar
  16. 16.
    Rogers, C.A.: Existence Theorems in the Geometry of Numbers. Ann. Math. II. Ser.48, 994–1002 (1947)Google Scholar
  17. 17.
    Rush, J.A., Sloane, N.J.A.: An improvement to the Minkowski-Hlawka bound for packing superballs. Mathematika34, 8–18 (1987)Google Scholar
  18. 18.
    Rush, J.A.: A lower bound on packing density. Invent. Math.98, 499–509 (1989)Google Scholar
  19. 19.
    Rush, J.A.: Thin lattice coverings. (Submitted)Google Scholar
  20. 20.
    Rush, J.A.: Constructive packings of cross polytopes (Submitted)Google Scholar
  21. 21.
    Sloane, N.J.A.: Self-dual codes and lattices. In: Relations Between Combinatorics and Other Parts of Mathematics. Proc. Symp. Pure Math34, 273–308 (1979)Google Scholar
  22. 22.
    Sloane, N.J.A.: Recent bounds for codes, sphere packings and related problems obtained by linear programming and other methods. Contemp. Math.9, 153–185 (1982)Google Scholar
  23. 23.
    Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications. Berlin Heidelberg New York: Springer, Vol. I, 1985 and Vol. II, 1988Google Scholar
  24. 24.
    Widder, D.V.: The Laplace Transform. Princeton: Princeton University Press 1941Google Scholar

Copyright information

© Springer International 1991

Authors and Affiliations

  • N. D. Elkies
    • 1
  • A. M. Odlyzko
    • 2
  • J. A. Rush
    • 3
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.AT & T Bell LaboratoriesMurray HillUSA
  3. 3.Department of MathematicsUniversity of EdinburghEdinburghScotland, UK

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