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Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set \[\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}\] be covered by a set of circles the sum of whose radii is $\leq 2$?
Cartan proved this is true with $2$ replaced by $2e$, which was improved to $2.59$ by Pommerenke [Po61]. Pommerenke [Po59] proved that $2$ is achievable if the set is connected (in fact the entire set is covered by a single circle with radius $2$).