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 for integers is defined as follows. For a lattice in Euclidean space with unit covolume, i.e., let denote the least length of a nonzero element of. Then is the maximum of over all such lattices.
The square root in the definition of the Hermite constant is a matter of historical convention.
Alternatively, the Hermite constant can be defined as the square of the maximal systole of a flat -dimensional torus of unit volume.
Examples
The Hermite constant is known in dimensions 1–8 and 24.| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 24 | |
For, one has. This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.
Estimates
It is known thatA stronger estimate due to Hans Frederick Blichfeldt is
where is the gamma function.