Hermite constant

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A hexagonal lattice with unit covolume (the area of the quadrilateral is 1). Both arrows are minimum non-zero elements for n − 1 {\displaystyle n-1} {\displaystyle n-1} with length λ n = γ n = 2 / 3 {\textstyle \lambda _{n}={\sqrt {\gamma _{n}}}={\sqrt {2/{\sqrt {3}}}}} {\textstyle \lambda _{n}={\sqrt {\gamma _{n}}}={\sqrt {2/{\sqrt {3}}}}}.

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 γ n {\displaystyle \gamma _{n}} {\displaystyle \gamma _{n}} for integers n > 0 {\displaystyle n>0} {\displaystyle n>0} is defined as follows. For a lattice L {\displaystyle L} {\displaystyle L} in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}} with unit covolume, i.e. vol ⁡ ( R n / L ) = 1 {\displaystyle \operatorname {vol} (\mathbb {R} ^{n}/L)=1} {\displaystyle \operatorname {vol} (\mathbb {R} ^{n}/L)=1}, let λ 1 ( L ) {\displaystyle \lambda _{1}(L)} {\displaystyle \lambda _{1}(L)} denote the least length of a nonzero element of L {\displaystyle L} {\displaystyle L}. Then γ n {\displaystyle {\sqrt {\gamma _{n}}}} {\displaystyle {\sqrt {\gamma _{n}}}} is the maximum of λ 1 ( L ) {\displaystyle \lambda _{1}(L)} {\displaystyle \lambda _{1}(L)} over all such lattices L {\displaystyle L} {\displaystyle L}.

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

Alternatively, the Hermite constant γ n {\displaystyle \gamma _{n}} {\displaystyle \gamma _{n}} can be defined as the square of the maximal systole of a flat n {\displaystyle n} {\displaystyle n}-dimensional torus of unit volume.

Examples

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

n 1 2 3 4 5 6 7 8 24
γ n {\displaystyle \gamma _{n}} {\displaystyle \gamma _{n}} 1 {\displaystyle 1} {\displaystyle 1} 2 3 {\displaystyle {\frac {2}{\sqrt {3}}}} {\displaystyle {\frac {2}{\sqrt {3}}}} 2 3 {\displaystyle {\sqrt[{3}]{2}}} {\displaystyle {\sqrt[{3}]{2}}} 2 {\displaystyle {\sqrt {2}}} {\displaystyle {\sqrt {2}}} 8 5 {\displaystyle {\sqrt[{5}]{8}}} {\displaystyle {\sqrt[{5}]{8}}} 2 3 6 {\displaystyle {\frac {2}{\sqrt[{6}]{3}}}} {\displaystyle {\frac {2}{\sqrt[{6}]{3}}}} 64 7 {\displaystyle {\sqrt[{7}]{64}}} {\displaystyle {\sqrt[{7}]{64}}} 2 {\displaystyle 2} {\displaystyle 2} 4 {\displaystyle 4} {\displaystyle 4}

For n = 2 {\displaystyle n=2} {\displaystyle n=2}, one has γ 2 = 2 / 3 {\displaystyle \gamma _{2}=2/{\sqrt {3}}} {\displaystyle \gamma _{2}=2/{\sqrt {3}}}. 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]

γ n ≤ ( 4 3 ) n − 1 2 . {\displaystyle \gamma _{n}\leq \left({\frac {4}{3}}\right)^{\frac {n-1}{2}}.} {\displaystyle \gamma _{n}\leq \left({\frac {4}{3}}\right)^{\frac {n-1}{2}}.}

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

γ n ≤ ( 2 π ) Γ ( 2 + n 2 ) 2 n , {\displaystyle \gamma _{n}\leq \left({\frac {2}{\pi }}\right)\Gamma \left(2+{\frac {n}{2}}\right)^{\frac {2}{n}},} {\displaystyle \gamma _{n}\leq \left({\frac {2}{\pi }}\right)\Gamma \left(2+{\frac {n}{2}}\right)^{\frac {2}{n}},} where Γ ( x ) {\displaystyle \Gamma (x)} {\displaystyle \Gamma (x)} is the gamma function.

See also

References

  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