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 γn 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 γn 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 γn 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. 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 Γ(x) is the gamma function.

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. JFM 55.0721.01. doi:10.1007/bf01454863.
4. ^ Kitaoka (1993) p. 42