Английская Википедия:Hermite constant

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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 γ_n 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 γ_n 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.

Estimates

It is known that

<math>\gamma_n \le \left( \frac 4 3 \right)^\frac{n-1}{2}.</math>

A stronger estimate due to Hans Frederick Blichfeldt is

<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

• Loewner's torus inequality

References

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book