| dbo:abstract |
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 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. In principle one can find explicit examples: for example, even just picking a few "random" lattices will work with high probability. The problem is that testing these lattices to see if they are solutions requires finding their shortest vectors, and the number of cases to check grows very fast with the dimension, so this could take a very long time. This result was stated without proof by Hermann Minkowski and proved by Edmund Hlawka. The result is related to a linear lower bound for the Hermite constant. (en) |
| dbo:wikiPageExternalLink |
https://archive.org/details/spherepackingsla0000conw_b8u0 https://pdfs.semanticscholar.org/c26b/5214362a547b8dd6f7ba3c8f32fad34b8faa.pdf https://web.archive.org/web/20200226042136/https:/pdfs.semanticscholar.org/c26b/5214362a547b8dd6f7ba3c8f32fad34b8faa.pdf |
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dbr:Riemann_zeta_function dbr:Mathematics dbr:Lower_bound dbr:Lattice_(group) dbc:Theorems_in_geometry dbr:Euclidean_space dbr:Packing_density dbr:Hermite_constant dbc:Hermann_Minkowski dbr:Kepler_conjecture dbc:Geometry_of_numbers dbr:Hypersphere dbr:Lattice_packing dbr:Lattice_point |
| dbp:authorlink |
Hermann Minkowski (en) Edmund Hlawka (en) |
| dbp:first |
Hermann (en) Edmund (en) |
| dbp:last |
Minkowski (en) Hlawka (en) |
| dbp:loc |
pages 265–276 (en) |
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1911 (xsd:integer) 1943 (xsd:integer) |
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dbc:Theorems_in_geometry dbc:Hermann_Minkowski dbc:Geometry_of_numbers |
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yago:WikicatTheoremsInGeometry yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293 |
| rdfs:comment |
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 This result was stated without proof by Hermann Minkowski and proved by Edmund Hlawka. The result is related to a linear lower bound for the Hermite constant. (en) |
| rdfs:label |
Minkowski–Hlawka theorem (en) |
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