Rado's covering problem for cubes and balls: a semi-survey

What is the largest constant $c\in [0,1]$ with the property that every finite collection $\mathcal{C}$ of axis-parallel squares in the plane admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $c$ of the area covered by $\mathcal{C}$? This problem was first raised by T.~Radó in 1928, who was motivated by a classical covering lemma in real analysis due to Vitali. R.~Rado later generalized the problem from axis-parallel squares in the plane to homothetic copies of any given convex body $K$ in ${\mathbb{R}}^d$, where now we are looking for an optimal constant $F(K)$. Our utmost interest is for cubes and balls in the high-dimensional regime $d\to \mathrm{\infty }$. The estimates that we currently have for cubes are much more precise than those for balls: namely if $Q^d$ is a $d$-dimensional cube, then $$(e^{-1}+o(1))\frac{2^{-d}}{d\log d}\le F(Q^d)\le 2^{-d},$$ while denoting $B^d$ a $d$-dimensional Euclidean ball, then $$(1+{\epsilon }_d)3^{-d}\le F(B^d)\le {2.447}^{-d},$$ where ${\epsilon }_d>0$ vanishes exponentially fast as $d\to \mathrm{\infty }$. The latter upper bound is obtained here by using the Kabatiansky–Levenshtein bound for the sphere packing problem.