Hermite constant explained

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

for integers is defined as follows. For a lattice in Euclidean space with unit covolume, i.e. , let denote the least length of a nonzero element of . Then is the maximum of over all such lattices .

The square root in the definition of the Hermite constant is a matter of historical convention.

Alternatively, the Hermite constant

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

Examples

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

scope=row		1	2	3	4	5	6	7	8	24
scope=row		1	4 3	2	4	8	64 3	64	28	424

For

, one has . This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.^[1]

Estimates

It is known that^[2]

$$\backslash gamma\_n\; \backslash le\; \backslash left(\backslash frac\; 4\; 3\; \backslash right)^\backslash frac.$$

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

$$\backslash gamma\_n\; \backslash le\; \backslash left(\backslash frac\; 2\; \backslash pi\; \backslash right)\backslash Gamma\backslash left(2\; +\; \backslash frac\; n\; 2\backslash right)^\backslash frac,$$where

is the gamma function.

See also

• Loewner's torus inequality
• Minkowski's theorem

References

• Book: Cassels, J.W.S. . J. W. S. Cassels

. J. W. S. Cassels . An Introduction to the Geometry of Numbers . Classics in Mathematics . . Reprint of 1971 . 1997 . 978-3-540-61788-4 .

• Book: Kitaoka, Yoshiyuki . Arithmetic of quadratic forms . Cambridge Tracts in Mathematics . 106 . Cambridge University Press . 1993 . 0-521-40475-4 . 0785.11021 .
• Book: Schmidt, Wolfgang M. . Wolfgang M. Schmidt

. Wolfgang M. Schmidt . Diophantine approximations and Diophantine equations . Lecture Notes in Mathematics . 1467 . . 1996 . 2nd . 3-540-54058-X . 0754.11020 . 9 .

Notes and References

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