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.