Hermite constant

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 ${\textstyle \lambda _{n}={\sqrt {\gamma _{n}}}={\sqrt {2/{\sqrt {3}}}}}$.

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.

Examples

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

References

1. Cassels (1971) p. 36
2. Leon Mächler; David Naccache (2022). "A Conjecture on Hermite Constants". Cryptology ePrint Archive.
3. Kitaoka (1993) p. 36
4. Blichfeldt, H. F. (1929). "The minimum value of quadratic forms, and the closest packing of spheres". Math. Ann. 101: 605–608. doi:10.1007/bf01454863. JFM   55.0721.01. S2CID   123648492.
5. Kitaoka (1993) p. 42

• Cassels, J.W.S. (1997). An Introduction to the Geometry of Numbers. Classics in Mathematics (Reprint of 1971 ed.). Springer-Verlag. ISBN   978-3-540-61788-4.
• Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. ISBN   0-521-40475-4. Zbl   0785.11021.
• Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. p. 9. ISBN   3-540-54058-X. Zbl   0754.11020.