Hermite constant

In mathematics, the Hermite constant, named after Charles Hermite, determines how short an element of a lattice in Euclidean space can be.

The constant $\gamma_n$ for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then $\sqrt{\gamma_n}$ 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. With the definition as stated, it turns out that the Hermite constant grows linearly in n as n becomes unbounded.

Alternatively, the Hermite constant $\gamma_n$ 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. For n = 2, one has $\gamma_2 = \tfrac{2}{\sqrt{3}}$. This value is attained by the hexagonal lattice of the Eisenstein integers.[1]

Estimates

It is known that[2]

$\gamma_n \le \left( \frac 4 3 \right)^{(n-1)/2}.$

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

$\gamma_n \le \left( \frac 2 \pi \right)\Gamma\left(2 + \frac n 2\right)^{2/n}.$