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}} for integers n > 0 {\displaystyle n>0} is defined as follows. For a lattice L {\displaystyle L} in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} with unit covolume, i.e. vol ( R n / L ) = 1 {\displaystyle \operatorname {vol} (\mathbb {R} ^{n}/L)=1} , let λ 1 ( L ) {\displaystyle \lambda _{1}(L)} denote the least length of a nonzero element of L {\displaystyle L} . Then γ n {\displaystyle {\sqrt {\gamma _{n}}}} is the maximum of λ 1 ( L ) {\displaystyle \lambda _{1}(L)} over all such lattices 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}} can be defined as the square of the maximal systole of a flat n {\displaystyle n} -dimensional torus of unit volume.
Examples The Hermite constant is known in dimensions 1–8 and 24.
For n = 2 {\displaystyle n=2} , one has γ 2 = 2 / 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.
Estimates It is known that
γ n ≤ ( 4 3 ) n − 1 2 . {\displaystyle \gamma _{n}\leq \left({\frac {4}{3}}\right)^{\frac {n-1}{2}}.}
A stronger estimate due to Hans Frederick Blichfeldt is
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}},}
where Γ ( x ) {\displaystyle \Gamma (x)} is the gamma function.
See also Loewner's torus inequality Minkowski's theorem
References
Cassels, J.W.S. (1997). An Introduction to the Geometry of Numbers. Classics in Mathematics (Reprint of 1971 ed.). Springer-Verlag. ISBN 978-3-540-61788-4. Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021. Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. p. 9. ISBN 3-540-54058-X. Zbl 0754.11020.