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 $d\rightarrow \infty$. 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 \[ (e^{-1}+o(1))\frac{2^{-d}}{d \log{d}} \leq F(Q^d)\leq 2^{-d}, \] while denoting $B^d$ a $d$-dimensional Euclidean ball, then \[ (1+ε_d)3^{-d}\leq F(B^d)\leq 2.447^{-d}, \] where $ε_d>0$ vanishes exponentially fast as $d\rightarrow \infty$. The latter upper bound is obtained here by using the Kabatiansky--Levenshtein bound for the sphere packing problem.

翻译：是否存在最大常数$c\in [0,1]$，使得对平面中任意由轴平行正方形构成的有限集合$\mathcal{C}$，总存在其不交子集$\mathcal{S}$，其覆盖面积至少占$\mathcal{C}$所覆盖面积的$c$倍？该问题由T. Radó于1928年首次提出，其动机源于Vitali在实分析中的经典覆盖引理。R. Rado后来将问题从平面中的轴平行正方形推广至$\mathbb{R}^d$中任意给定凸体$K$的位似复制，此时我们寻求最优常数$F(K)$。我们最关注高维情形$d\rightarrow \infty$下的立方体与球。目前对立方体的估计远精确于对球的估计：若$Q^d$为$d$维立方体，则\[ (e^{-1}+o(1))\frac{2^{-d}}{d \log{d}} \leq F(Q^d)\leq 2^{-d}, \]而记$B^d$为$d$维欧氏球，则\[ (1+ε_d)3^{-d}\leq F(B^d)\leq 2.447^{-d}, \]其中$ε_d>0$随$d\rightarrow \infty$指数衰减。本文通过利用球堆积问题的Kabatiansky-Levenshtein界得到后者的上界。