Hermite constant
# 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 $γ_n$ for integers n > 0 is defined as follows. For a lattice L in Euclidean space $R^n$ with unit covolume, i.e. $vol(R^n/L)$ = 1, let $λ_1(L)$ denote the least length of a nonzero element of L. Then $√γ_n$ is the maximum of $λ_1(L)$ 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 $γ_n$ 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.

| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 24 |
| - | - | - | - | - | - | - | - | - | - |
| $\gamma _{n}^{n}$ | $1$ | ${\frac {4}{3}}$ | $2$ | $4$ | $8$ | ${\frac {64}{3}}$ | $64$ | $2^{8}$ | $4^{24}$ |

For n = 2, one has $γ_2$ = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers.

## Estimates

It is known that

    $\gamma _{n}\leq \left({\frac {4}{3}}\right)^{\frac {n-1}{2}}.$

A stronger estimate due to Hans Frederick Blichfeldt is

    $\gamma _{n}\leq \left({\frac {2}{\pi }}\right)\Gamma \left(2+{\frac {n}{2}}\right)^{\frac {2}{n}},$

where $\Gamma (x)$ is the gamma function.

