Hermite constant

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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 Rⁿ, let λ₁(L) denote the least length of a nonzero element of L. Then is the maximum of λ₁(L) over all lattices L of unit covolume, i.e. vol(Rⁿ/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.

[edit] Example

For n = 2, one has . This value is attained by the Eisenstein integers.

[edit] See also

• Loewner's torus inequality