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 γn 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 √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. 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.

Estimates

It is known that A stronger estimate due to Hans Frederick Blichfeldt is where \Gamma(x) is the gamma function.