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In [[mathematics]], the '''Minkowski–Hlawka theorem''' is a result on the [[lattice packing]] of [[hypersphere]]s in dimension ''n'' > 1. It states that there is a [[lattice (group)|lattice]] in [[Euclidean space]] of dimension ''n'', such that the corresponding best packing of hyperspheres with centres at the [[lattice point]]s has density Δ satisfying | |||
:<math>\Delta \geq \frac{\zeta(n)}{2^{n-1}},</math> | |||
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 [[Edmund Hlawka]] (1944). The result is related to a linear lower bound for the [[Hermite constant]]. | |||
==See also== | |||
*[[Kepler conjecture]] | |||
==References== | |||
*{{cite book | |||
| first = John H. | |||
| last = Conway | |||
| authorlink = John Horton Conway | |||
| coauthors = [[Neil Sloane|Neil J.A. Sloane]] | |||
| year = 1999 | |||
| title = Sphere Packings, Lattices and Groups | |||
| edition = 3rd ed. | |||
| publisher = Springer-Verlag | |||
| location = New York | |||
| isbn = 0-387-98585-9 | |||
}} | |||
{{DEFAULTSORT:Minkowski-Hlawka theorem}} | |||
[[Category:Geometry of numbers]] | |||
[[Category:Theorems in geometry]] | |||
Revision as of 03:55, 23 January 2014
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 Edmund Hlawka (1944). The result is related to a linear lower bound for the Hermite constant.
See also
References
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