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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
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:<math>\Delta \geq \frac{\zeta(n)}{2^{n-1}},</math>
 
with &zeta; the [[Riemann zeta function]]. Here as ''n'' &rarr; &infin;, &zeta;(''n'') &rarr; 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 19:47, 21 February 2014

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