Why can we cover $\mathbb R^N$ with open balls of radius $r$ such that each point is in at most $N + 1$ balls?
Solution 1:
user125932 mentioned in a comment that this seems to be an open problem, since it would imply that the covering density of any $n$-dimensional ball is at most $n+1$. As of 2018, it still seems that nobody can prove a better upper bound on that covering density than $Θ( n · \log n )$; see here and here. In particular, the first linked paper explicitly states that as $n→∞$, unit balls can cover $\mathbb{R}^n$ with density $\big(\frac12+o(1)\big)\ n\ln n$, as Corollary 2.