Hermite constant

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In mathematics, the Hermite constant, named after Charles Hermite, determines how short an element of a lattice in Euclidean space can be.

The constant \gamma_n for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn unit covolume, i.e. vol(Rn/L) = 1, 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 such lattices L.

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

Example[edit]

The Hermite constant is known in dimensions 1–8 and 24. For n = 2, one has \gamma_2 = \tfrac{2}{\sqrt{3}}. This value is attained by the hexagonal lattice of the Eisenstein integers.[1]

Estimates[edit]

It is known that[2]

\gamma_n \le (4/3)^{(n-1)/2} \ .

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

\gamma_n \le (2/\pi)\Gamma(2+n/2)^{2/n} \ .

See also[edit]

References[edit]

  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. JFM 55.0721.01. 
  4. ^ Kitaoka (1993) p.42