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.Cassels (1971) p. 36
Estimates It is known thatKitaoka (1993) p. 36
A stronger estimate due to Hans Frederick Blichfeldt isKitaoka (1993) p. 42
where is the gamma function.
See also Loewner's torus inequality Minkowski's theorem
References