Hermite constant

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 √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.

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 = √γ_n = ${\displaystyle {\sqrt {2/{\sqrt {3}}}}}$

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

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

For n = 2, one has γ₂ = ⁠2/√3⁠. This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.^[1]

The constants for the missing n values are conjectured.^[2]

Estimates

It is known that^[3]

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

A stronger estimate due to Hans Frederick Blichfeldt^[4] is^[5]

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

where ${\displaystyle \Gamma (x)}$ 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.