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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).

What about $n>8$?

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The only other case that is known is n=24: Abhinav Kumar and I solved that case (Annals of Mathematics 170 (2009), 1003-1050, http://front.math.ucdavis.edu/math.MG/0403263).

It might be possible to do n=9 using known methods, but it would be an enormous calculation. For the status as of a few years ago, see the end of Mathieu Dutour Sikiric, Achill Schuermann, and Frank Vallentin's paper "Classification of eight dimensional perfect forms" (http://front.math.ucdavis.edu/0609.5388).

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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 $n>9$? –  Portland Mar 16 '11 at 22:00
    
Getting bounds (upper or lower) on the optimal lattice packing density is definitely an active topic of research. Finding exact answers is trickier, since there aren't many good opportunities. I don't know whether anyone is actively working on n=9, but maybe it will be solved in the next decade. I don't think there's much likelihood of reaching another non-consecutive value of n (like n=24), but of course it is difficult to predict. I hope n=10 will be doable someday, since that seems to be the first case in which lattices are inferior to nonlattice packings. –  Henry Cohn Mar 17 '11 at 3:39
    
I imagine that before n gets too large, the problem will become utterly intractable. However, this could be overly pessimistic. Incidentally, there are also lots of interesting questions for other rings (say, lattices with Gaussian, Eisenstein, or Hurwitz structures). See, for example, front.math.ucdavis.edu/0901.1587 and front.math.ucdavis.edu/0810.2336. –  Henry Cohn Mar 17 '11 at 3:44
    
Thank you very much again Henry. –  Portland Mar 17 '11 at 12:10
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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}$.

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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. –  DavidLHarden Mar 16 '11 at 5:32
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