In
mathematics, the Hermite constant, named after
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
, determines how short an element of a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
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 of a flat ''n''-dimensional
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
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 or 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 angle of 120° ...
of the
Eisenstein integers
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
:z = a + b\omega ,
where and are integers and
:\omega = \f ...
.
Estimates
It is known that
[Kitaoka (1993) p. 36]
:
A stronger estimate due to
Hans Frederick Blichfeldt is
[Kitaoka (1993) p. 42]
:
where
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.
See also
*
Loewner's torus inequality
In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.
Statement
In 1949 Charles Loewner proved that every metric on ...
References
*
*
*
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