Hermite constant

Template: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 Template: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

A hexagonal lattice with unit covolume (the area of the quadrilateral is 1). Both arrows are minimum non-zero elements for n = 2 with length λ_n = Template:Sqrt = $\sqrt{2/\sqrt{3}}$

The Hermite constant is known in dimensions 1–8 and 24.

Template:Math	1	2	3	4	5	6	7	8	24
Template:Math	$1$	$\frac{4}{3}$	$2$	$4$	$8$	$\frac{64}{3}$	$64$	$2^8$	$4^{24}$

For n = 2, one has γ₂ = Template:Sfrac. This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.

The constants for the missing Template:Math values are conjectured.

Estimates

It is known that

${\gamma }_n\le {\left(\frac{4}{3}\right)}^{\frac{n-1}{2}}.$

A stronger estimate due to Hans Frederick Blichfeldt is

${\gamma }_n\le \left(\frac{2}{\pi }\right)\mathrm{\Gamma }{(2+\frac{n}{2})}^{\frac{2}{n}},$

where $\mathrm{\Gamma }(x)$ is the gamma function.

See also

• Loewner's torus inequality

References

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