Hermite constant

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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 γn 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 γn 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 γn can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.

Example

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

n 1 2 3 4 5 6 7 8 24
                   

For n = 2, one has γ2 = 2/3. This value is attained by the hexagonal lattice of the Eisenstein integers.[1]

Estimates

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It is known that[2]

 

A stronger estimate due to Hans Frederick Blichfeldt[3] is[4]

 

where   is the gamma function.

See also

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References

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  1. ^ Cassels (1971) p. 36
  2. ^ Kitaoka (1993) p. 36
  3. ^ Blichfeldt, H. F. (1929). "The minimum value of quadratic forms, and the closest packing of spheres". Math. Ann. 101: 605–608. doi:10.1007/bf01454863. JFM 55.0721.01. S2CID 123648492.
  4. ^ Kitaoka (1993) p. 42