Definition:Hermite Constant

Definition

Let $n \in \N$ be a natural number.

Let $L$ be an integer lattice in Euclidean space $R^n$ unit covolume.

That is:

$\map {\operatorname {vol} } {\dfrac {R^n} L} = 1$

Let $\map {\lambda_1} L$ denote the least length of a nonzero element of $L$.

Let $\sqrt {\gamma_n}$ be the maximum of $\map {\lambda_1} L$ over all such integer lattices $L$.

The Hermite constant of dimension $n$ is the constant $\gamma_n$.

Source of Name

This entry was named for Charles Hermite.

Historical Note

The square root sign $\sqrt {\, \cdot \,}$ in the definition of the Hermite constants is a historical convention.

Sources

• 1929: H.F. Blichfeldt: The minimum value of quadratic forms, and the closest packing of spheres (Math. Ann. Vol. 101: pp. 605 – 608)

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,33333 33333 33 \ldots$

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