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 Rⁿ with unit covolume, i.e. vol(Rⁿ/L) = 1, let λ₁(L) denote the least length of a nonzero element of L. Then √γ_n 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 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
${\displaystyle \gamma _{n}^{n))$	${\displaystyle 1}$	${\displaystyle {\frac {4}{3))}$	${\displaystyle 2}$	${\displaystyle 4}$	${\displaystyle 8}$	${\displaystyle {\frac {64}{3))}$	${\displaystyle 64}$	${\displaystyle 2^{8))$	${\displaystyle 4^{24))$

For n = 2, one has γ₂ = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers.^[1]

Estimates

It is known that^[2]

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

A stronger estimate due to Hans Frederick Blichfeldt^[3] is^[4]

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

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