- Minkowski-Hlawka theorem
In
mathematics , the Minkowski-Hlawka theorem is a result on thelattice packing ofhypersphere s in dimension "n" > 1. It states that there is a lattice inEuclidean space of dimension "n", such that the corresponding best packing of hyperspheres with centres at thelattice point s 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 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 mathematicianEdmund Hlawka (1944).References
*cite book
first = John H.
last = Conway
authorlink = John Horton Conway
coauthors = Neil J.A. Sloane
year = 1999
title = Sphere Packings, Lattices and Groups
edition = 3rd ed.
publisher = Springer-Verlag
location = New York
id = ISBN 0-387-98585-9
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