Minkowski–Hlawka theorem

In mathematics, the Minkowski–Hlawka theorem is a result on the lattice packing of hyperspheres in dimension n > 1. It states that there is a lattice in Euclidean space of dimension n, such that the corresponding best packing of hyperspheres with centres at the lattice points has density Δ satisfying

\Delta \geq \frac{\zeta(n)}{2^{n-1}},

with ζ the Riemann zeta function. Here as n → ∞, ζ(n) → 1. The proof of this theorem is indirect and does not give an explicit example, however, and there is still no known simple and explicit way to construct lattices with packing densities exceeding this bound for arbitrary n. (One can in principle always find examples by brute force search, but the number of cases to check grows very fast with the dimension, so this could take a very long time.)

This is a result of Hermann Minkowski (1911, pages 265–276) and Edmund Hlawka (1943). The result is related to a linear lower bound for the Hermite constant.

References

Conway, John H.; Neil J.A. Sloane (1999). Sphere Packings, Lattices and Groups (3rd ed. ed.). New York: Springer-Verlag. ISBN 0-387-98585-9.
Hlawka, Edmund (1943), "Zur Geometrie der Zahlen", Math. Z. 49: 285–312, doi:10.1007/BF01174201, MR 0009782
Minkowski (1911), Gesammelte Abhandlungen 1, Leipzig: Teubner

Minkowski–Hlawka theorem

See also

References