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.
Example
A hexagonal lattice with unit covolume (the area of the quadrilateral is 1). Both arrows are minimum non-zero elements for with length .
The Hermite constant is known in dimensions 1–8 and 24.