Hermite Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

The Hermite constant is defined for Dimension $n$ as the value

$$\gamma_n={\sup_f \min_{x_i} f(x_1, x_2, \ldots, x_n)\over [\hbox{discriminant}(f)]^{1/n}}$$

(Le Lionnais 1983). In other words, they are given by

$$\gamma_n=4\left({\delta_n\over V_n}\right)^{2/n},$$

where $\delta_n$ is the maximum lattice Packing Density for Hypersphere Packing and $V_n$ is the Content of the $n$-Hypersphere. The first few values of $(\gamma_n)^n$ are 1, 4/3, 2, 4, 8, 64/3, 64, 256, .... Values for larger $n$ are not known.

For sufficiently large $n$,

$${1\over 2\pi e}\leq {\gamma_n\over n}\leq {1.744\ldots\over 2\pi e}.$$

See also Hypersphere Packing, Kissing Number, Sphere Packing

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/hermit/hermit.html

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 20, 1993.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 38, 1983.