# Extremal lattices

Denote by $$\mu_n$$ the largest value such that there exists a lattice of determinant $$1$$ in $$\mathbb R^n$$ for which the distances between different lattice points are greater or equal to $$\mu_n$$.

Korkine, Zolotareff, and then Blichfeldt found $$\mu_n$$ for $$n=2,\ldots , 8$$ (cf. H.F. Blichfeldt, The minimum value of positive quadratic forms in six, seven, and eight variables. Math. Z. 39 (1935), 1-15, EuDML).

What about $$n>8$$?

• Thanks Henry, very useful. Is it still an active topic of research? Do you believe there is "reasonable hope" to crack the problem for other values of $n>9$? Mar 16, 2011 at 22:00
This is also known as the problem of finding the best lattice packing of spheres in $n$ dimensons. The density of the best lattice packing in $n$ dimensions is, following your notation, $\frac{ \pi^{n/2} }{ \Gamma( 1 + \frac{n}{2}) }\mu_{n}^{n}$.
• I'm sorry. That should be $\frac{ \pi^{n/2} }{ \Gamma( 1 + \frac{n}{2}) }\frac{\mu_{n}^{n}}{2^{n}}$, since $\mu_{n}$ is the diameter of the spheres in the packing, not their radius. Mar 16, 2011 at 5:32