Английская Википедия:Hermite constant
Шаблон:Short description In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be.
The constant γn for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then Шаблон:Sqrt is the maximum of λ1(L) over all such lattices L.
The square root in the definition of the Hermite constant is a matter of historical convention.
Alternatively, the Hermite constant γn can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.
Example
The Hermite constant is known in dimensions 1–8 and 24.
Шаблон:Math | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 24 |
---|---|---|---|---|---|---|---|---|---|
Шаблон:Math | <math>1</math> | <math>\frac 4 3</math> | <math>2</math> | <math>4</math> | <math>8</math> | <math>\frac {64} 3</math> | <math>64</math> | <math>2^8</math> | <math>4^{24}</math> |
For n = 2, one has γ2 = Шаблон:Sfrac. This value is attained by the hexagonal lattice of the Eisenstein integers.[1]
Estimates
It is known that[2]
- <math>\gamma_n \le \left( \frac 4 3 \right)^\frac{n-1}{2}.</math>
A stronger estimate due to Hans Frederick Blichfeldt[3] is[4]
- <math>\gamma_n \le \left( \frac 2 \pi \right)\Gamma\left(2 + \frac n 2\right)^\frac{2}{n},</math>
where <math>\Gamma(x)</math> is the gamma function.
See also
References
Шаблон:Systolic geometry navbox
- ↑ Cassels (1971) p. 36
- ↑ Kitaoka (1993) p. 36
- ↑ Шаблон:Cite journal
- ↑ Kitaoka (1993) p. 42