Hermite constant

In mathematics, the Hermite constant, named after Charles Hermite, determines how short an 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 unit covolume, i.e. vol(Rⁿ/L) = 1, let λ₁(L) denote the least length of a nonzero element of L. Then √γ_n 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. With the definition as stated, it turns out that the Hermite constant grows linearly in n.

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.

n	1	2	3	4	5	6	7	8	24
$\gamma _{n}^{n}$	$1$	${\frac {4}{3}}$	$2$	$4$	$8$	${\frac {64}{3}}$	$64$	$2^{8}$	$4^{24}$

For n = 2, one has γ₂ = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers.

Estimates

It is known that

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

A stronger estimate due to Hans Frederick Blichfeldt is

$\gamma _{n}\leq \left({\frac {2}{\pi }}\right)\Gamma \left(2+{\frac {n}{2}}\right)^{\frac {2}{n}},$

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

See also

• Loewner's torus inequality

References

• Cassels, J.W.S. (1997). An Introduction to the Geometry of Numbers. Classics in Mathematics (Reprint of 1971 ed.). Springer-Verlag. ISBN 978-3-540-61788-4.
• Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021.
• Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. p. 9. ISBN 3-540-54058-X. Zbl 0754.11020.