Hermite constant

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


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