A sharp covering theorem and Solyanik estimates for Euclidean balls

For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<δ<1/2$, we select a subfamily whose $(1+δ)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_nδ^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{