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 γ_n for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rⁿ with unit covolume, i.e. vol(Rⁿ/L) = 1, let λ₁(L) denote the least length of a nonzero element of L. Then √γ_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.

Alternatively, the Hermite constant γ_n can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.

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

For n = 2, one has γ₂ =

⁠2/√3⁠. This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.

The constants for the missing n values are conjectured.

It is known that

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

A stronger estimate due to Hans Frederick Blichfeldt is

${\displaystyle \gamma _{n}\leq \left({\frac {2}{\pi }}\right)\Gamma \left(2+{\frac {n}{2}}\right)^{\frac {2}{n}},}$

where ${\displaystyle \Gamma (x)}$is the gamma function.

• Loewner's torus inequality

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• 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.