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.

Example

A hexagonal lattice with unit covolume (the area of the quadrilateral is 1). Both arrows are minimum non-zero elements for ${\displaystyle n-1}$ with length ${\textstyle \lambda _{n}={\sqrt {\gamma _{n}}}={\sqrt {2/{\sqrt {3}}}}}$.

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 ${\displaystyle n=2}$, one has ${\displaystyle \gamma _{2}=2/{\sqrt {3}}}$. This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.^[1]

The constants for the missing ${\displaystyle n}$ values are conjectured.^[2]

Estimates

It is known that^[3]

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

A stronger estimate due to Hans Frederick Blichfeldt^[4] is^[5]

$${\displaystyle \gamma _{n}\leq \left({\frac {2}{\pi }}\right)\Gamma \left(2+{\frac {n}{2}}\right)^{\frac {2}{n}},}$$ where ${\displaystyle \Gamma (x)}$ is the gamma function.

See also

• Loewner's torus inequality
• Minkowski's theorem