It is a famous problem what is the densest (or, most efficient) packing of unit balls in Euclidean n-dimensional spaces and, of course, you can ask the same questions for other convex bodies. A sort of a dual problem, also famous, is the question of the most efficient covering the n-dimensional Euclidean space by balls or by translates of other convex bodies.
A very basic insight is that covering is much more efficient than packing. Can you explain the reason for that? The only thing that came up to my mind was that convex sets are roundish from the outside but not from the inside which makes it easier to cover with them.
But how efficiently can we cover the n-dimensional Euclidean space with translates of a convex body $K$?
C.A. Rogers with coauthors had some very fundamental results regarding covering (and packing) in the 50s and 60s. Or Ordentlich, Oded Regev and Barak Weiss have made a terrific improvement for one of the main questions in this area. They showed that for every convex set K, you can find a lattice L of determinant 1, and r>0 such that and
The best earlier upper bound was .
One surprising aspect of the proof is the use of finite field Kakeya-type questions. Since the breakthrough by Zeev Dvir, many people had hopes for applications from the finite fields results to the Euclidean setting (in particular, to the Euclidean Kakeya problem itself) and it is rare to see such applications. The proof here relies on results by Dvir, Kopparty, Saraf, Sudan, and Lev. The Ordentlich-Regev-Weiss paper is not yet on the arXiv but the lecture gives a good picture about the results and the proofs.
The definition of the covering density of L with respect to a convex body K
The definition of the covering density of K
Old results for the Euclidean ball
Old results for general convex bodies
The first main result
The required finite field Kakeya theorem.