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

For n = 2, one has γ₂ = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers.[1]

Estimates

It is known that[2]

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

A stronger estimate due to Hans Frederick Blichfeldt[3] is[4]

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

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

See also

• Loewner's torus inequality

References

1. Cassels (1971) p. 36
2. Kitaoka (1993) p. 36
3. Blichfeldt, H. F. (1929). "The minimum value of quadratic forms, and the closest packing of spheres". Math. Ann. 101: 605–608. doi:10.1007/bf01454863. JFM 55.0721.01.
4. Kitaoka (1993) p. 42

• 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. 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. 1467 (2nd ed.). Springer-Verlag. p. 9. ISBN 3-540-54058-X. Zbl 0754.11020.