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 ${\displaystyle \gamma _{n}}$ for integers ${\displaystyle n>0}$ is defined as follows. For a lattice ${\displaystyle L}$ in Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with unit covolume, i.e. ${\displaystyle \operatorname {vol} (\mathbb {R} ^{n}/L)=1}$, let ${\displaystyle \lambda _{1}(L)}$ denote the least length of a nonzero element of ${\displaystyle L}$. Then ${\displaystyle {\sqrt {\gamma _{n}}}}$ is the maximum of ${\displaystyle \lambda _{1}(L)}$ over all such lattices ${\displaystyle L}$.

The square root in the definition of the Hermite constant is a matter of historical convention.

Alternatively, the Hermite constant ${\displaystyle \gamma _{n}}$ can be defined as the square of the maximal systole of a flat ${\displaystyle n}$-dimensional torus of unit volume.