Hermite constant

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

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 ${\gamma }_n$ for integers $n>0$ is defined as follows. For a lattice $L$ in Euclidean space ${ℝ}^n$ with unit covolume, i.e. $\mathrm{vol}({ℝ}^n/L)=1$, let ${\lambda }_1(L)$ denote the least length of a nonzero element of $L$. Then $\sqrt{{\gamma }_n}$ is the maximum of ${\lambda }_1(L)$ 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 ${\gamma }_n$ can be defined as the square of the maximal systole of a flat $n$-dimensional torus of unit volume.

Examples

The Hermite constant is known in dimensions 1–8 and 24.

Template:Math	1	2	3	4	5	6	7	8	24
Template:Math	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt[3]{2}$	$\sqrt{2}$	$\sqrt[5]{8}$	$\frac{2}{\sqrt[6]{3}}$	$\sqrt[7]{64}$	$2$	$4$

For $n=2$, one has ${\gamma }_2=2/\sqrt{3}$. This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.

Estimates

It is known that

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

A stronger estimate due to Hans Frederick Blichfeldt is

$${\gamma }_n\le \left(\frac{2}{\pi }\right)\mathrm{\Gamma }{(2+\frac{n}{2})}^{\frac{2}{n}},$$ where $\mathrm{\Gamma }(x)$ is the gamma function.

See also

• Loewner's torus inequality
• Minkowski's theorem

References

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