mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

can be. The constant ''γ_n'' for integers ''n'' > 0 is defined as follows. For a lattice ''L'' in Euclidean space R^''n'' with unit covolume, i.e. vol(R^''n''/''L'') = 1, let ''λ''₁(''L'') denote the least length of a nonzero element of ''L''. Then is the maximum of ''λ''₁(''L'') over all such lattices ''L''. The

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

in the definition of the Hermite constant is a matter of historical convention. Alternatively, the Hermite constant ''γ_n'' can be defined as the square of the maximal

systole Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. Its contrasting phase is diastole, the relaxed phase of the cardiac cycle when the chambers of the heart are refilling ...

of a flat ''n''-dimensional

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

of unit volume.

Example

The Hermite constant is known in dimensions 1–8 and 24. For ''n'' = 2, one has ''γ''₂ = . This value is attained by the

hexagonal lattice The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an ...

of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area. The constants for the missing values are conjectured.

Estimates

It is known thatKitaoka (1993) p. 36 :

\gamma_n \le \left( \frac 4 3 \right)^\frac.

A stronger estimate due to Hans Frederick Blichfeldt isKitaoka (1993) p. 42 :

\gamma_n \le \left( \frac 2 \pi \right)\Gamma\left(2 + \frac n 2\right)^\frac,

where

\Gamma(x)

is the

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

References

* * * {{Systolic geometry navbox Systolic geometry Geometry of numbers Mathematical constants

Example

Estimates

See also

References