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Hermite Constants

The Hermite constant in dimension n>=1 is

gamma_n=sup_(A)(min_(z in Z^n\{0})z^TAz)/((detA)^(1/n)),
(1)

where A ranges over real symmetric matrices of size n×n that are positive definite and the minimum is over nonzero integer vectors. Equivalently, it is given by

gamma_n=4((delta_n)/(V_n))^(2/n),
(2)

where delta_n is the maximum lattice packing density for hypersphere packing and V_n is the content of the n-dimensional unit ball. The first few values of (gamma_n)^n are 1, 4/3, 2, 4, 8, 64/3, 64, 256, ... (OEIS A007361 and A007362; Gruber and Lekkerkerker 1987, p. 518). The value gamma_(24)=4 is also known (Cohn and Kumar 2009).

For sufficiently large n,

1/(2pie)<=(gamma_n)/n<=(1.744)/(2pie)
(3)

(Finch 2003, p. 507).

See also

Discriminant, Hypersphere Packing, Kissing Number, Sphere Packing

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References

Cassels, J. W. S. An Introduction to the Geometry of Numbers, 2nd ed. New York: Springer-Verlag, p. 332, 1997.Cohn, H. and Kumar, A. "Optimality and Uniqueness of the Leech Lattice among Lattices." Ann. Math. 170, 1003-1050, 2009. https://doi.org/10.4007/annals.2009.170.1003.Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 20, 1993.Finch, S. R. "Hermite's Constants." §8.7 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 506-508, 2003.Gruber, P. M. and Lekkerkerker, C. G. Geometry of Numbers, 2nd ed. Amsterdam, Netherlands: North-Holland, 1987.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 38, 1983.Sloane, N. J. A. Sequences A007361/M3201 and A007362/M2209 in "The On-Line Encyclopedia of Integer Sequences."

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Hermite Constants

Cite this as:

Weisstein, Eric W. "Hermite Constants." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HermiteConstants.html

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