Minkowski-Hlawka theorem

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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

with ζ the Riemann zeta function. Here as n → ∞, ζ(n) → 1. The proof of this theorem is nonconstructive, however, and it is still not known how to construct lattices with packing densities exceeding this bound for arbitrary n.

This is a result of Hermann Minkowski (1905, not published) and the Austrian mathematician Edmund Hlawka (1944).

[edit] References

• Conway, John H.; Neil J.A. Sloane (1999). Sphere Packings, Lattices and Groups, 3rd ed., New York: Springer-Verlag. ISBN 0-387-98585-9.