Hermite constant

In mathematics, the Hermite constant γ_"n" for integers "n" > 0, named after Charles Hermite, is defined as follows. Given a lattice "L" in Euclidean space R^"n",let λ₁("L") denote the least length of a nonzero element of "L". Then $sqrt\{gamma\_n\}$ is the maximum of λ₁("L") over all lattices "L" of unit covolume, i.e. vol(R^"n"/"L") = 1.

The square root in the definition of the Hermite constant is a matter of historical convention. With the definition as stated, it turns out that the Hermite constant grows linearly in "n" as "n" becomes unbounded.

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.

Example

For "n" = 2, one has $scriptstylegamma\_2\; =\; frac\{2\}\{sqrt\{3$. This value is attained by the Eisenstein integers.

ee also

*Loewner's torus inequality