152:. 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.
148:) → 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
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is a star-shaped centrally symmetric body (such as a ball) containing less than 2 primitive lattice vectors then it contains no nonzero lattice vectors.
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The
Minkowski–Hlawka theorem follows easily from this, using the fact that if
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proved the following generalization of the
Minkowski–Hlawka theorem. If
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218:, and similarly the average number of primitive lattice vectors in
126:{\displaystyle \Delta \geq {\frac {\zeta (n)}{2^{n-1}}},}
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Existence theorem on the lattice packing of hyperspheres
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293:Hlawka, Edmund (1943), "Zur Geometrie der Zahlen",
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332:"A mean value theorem in geometry of numbers"
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278:(3rd ed.). New York: Springer-Verlag.
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155:This result was stated without proof by
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171:). The result is related to a linear
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274:Sphere Packings, Lattices and Groups
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50:> 1. It states that there is a
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325:, vol. 1, Leipzig: Teubner
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330:Siegel, Carl Ludwig (1945),
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144:→ ∞, ζ(
36:Minkowski–Hlawka theorem
18:Minkowski-Hlawka theorem
323:Gesammelte Abhandlungen
198:with Jordan volume vol(
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127:
157:Hermann Minkowski
138:Riemann zeta function
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404:Theorems in geometry
194:is a bounded set in
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399:Geometry of numbers
38:is a result on the
321:Minkowski (1911),
307:10.1007/BF01174201
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70:Δ satisfying
409:Hermann Minkowski
251:Kepler conjecture
165:Edmund Hlawka
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16:(Redirected from
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377:, archived from
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268:Neil J.A. Sloane
183:Siegel's theorem
177:Hermite constant
136:with ζ the
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44:hyperspheres
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301:: 285–312,
173:lower bound
32:mathematics
393:Categories
257:References
140:. Here as
375:124272126
111:−
90:ζ
84:≥
81:Δ
295:Math. Z.
270:(1999).
245:See also
175:for the
367:0012093
359:1969027
315:0009782
222:is vol(
206:is vol(
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68:density
52:lattice
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34:, the
382:(PDF)
371:S2CID
355:JSTOR
341:, 2,
335:(PDF)
280:ISBN
169:1943
161:1911
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347:doi
303:doi
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363:MR
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239:S
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