TITLE: Optimal sphere packings ==> Thinnest ball coverings?
QUESTION [29 upvotes]: It was proved by Kershner long ago that the thinnest (least density)
covering of the plane by congruent disks can be obtained
by enlarging the radii of the optimal circle packing to just cover
the gaps in that packing:

          


My question is:

Q. Is it known that, in the dimensions for which the optimal sphere packing
  is known $(2, 3, 8, 24)$, is it also known that a thinnest ball covering can be achieved
  by enlarging the spheres to just cover the interstices in the packing?

For example, now that the Kepler problem has been settled by Hales,
does it follow that a thinnest ball packing of $\mathbb{R}^3$ can be
obtained from the cannonball packing by enlarging the spheres to just cover
the gaps?

          



Kershner, Richard. "The number of circles covering a set." American Journal of Mathematics 61.3 (1939): 665-671.

REPLY [32 votes]: Nope, this is false in three dimensions (where the body-centered cubic beats the face-centered cubic for covering) and eight dimensions (where $E_8$ is not even locally optimal). The Leech lattice is locally optimal and may be the best covering in twenty-four dimensions, but that would be a special fact rather than an instance of a general pattern. Overall sphere covering is more complicated than sphere packing, and there seems to be no close relationship between the optimal solutions.