Hermite constant

 ( Systolic Geometry ) 

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

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

For $n=2$, one has $\backslash gamma\_2=2/\backslash sqrt\{3\}$. This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.Cassels (1971) p. 36

The constants for the missing $n$ values are conjectured.

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Estimates It is known thatKitaoka (1993) p. 36

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

A stronger estimate due to Hans Frederick Blichfeldt isKitaoka (1993) p. 42

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

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See also

• Loewner's torus inequality
• Minkowski's theorem

• J.W.S. Cassels (1997). 9783540617884, Springer-Verlag. . ISBN 9783540617884
• Yoshiyuki Kitaoka (1993). 9780521404754, Cambridge University Press. ISBN 9780521404754
• Wolfgang M. Schmidt (1996). 354054058X, Springer-Verlag. 354054058X

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