What's the current state of knowledge regarding packings of spheres in $n$-space that minimize the supremum of the sizes of the holes? This notion of tightness is more rigid than asymptotic density. I would expect tightest packings to coincide with densest periodic packings in low dimensions but not when $n$ is sufficiently large.
$\begingroup$
$\endgroup$
2
asked Sep 13, 2014 at 14:14
-
1$\begingroup$ Are you asking about the packing-covering problem (see e.g. arXiv:math/0412320)? $\endgroup$– Yoav KallusCommented Sep 13, 2014 at 14:32
-
$\begingroup$ Exactly! This answers my question. $\endgroup$– James ProppCommented Sep 13, 2014 at 16:53
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The problem you are asking about is sometimes known as the packing-covering problem, since it asks for a configuration with a fixed packing radius that minimizes the covering radius, irrespective of mean density. The lattice version of the problem is solved in some dimensions (see Table 3 of arXiv:math/0412320), and indeed the answer is different than the optimal packing lattice in dimensions $d>2$ and from the optimal covering lattice in dimensions $d>3$. A surprisingly open problem according to the paper linked is whether in some dimension, the optimal packing-covering lattice has a ratio greater or equal to 2 between the covering radius and the packing radius. That would mean that no lattice packing in that dimension can be saturated, since there must be a hole where another translate of the lattice can fit.
answered Sep 13, 2014 at 17:36