Area stability of plank covers

We prove a conjecture of András Bezdek on the stability of plank covers of the planar disk $D$. Namely, we show that if a sufficiently small concentric disk $rD$ is removed from $D$, then every finite family of planks covering the resulting annulus can be re-arranged to cover the whole disk. For the proof, we show a strong stability result for a related covering problem. If the hole has area $a\ge 0$ then the total overlap is $\ge ca\mathrm{/}r$ where $c>0$ is a constant. This overlap estimate is not specific to the disk and is applicable to all planar symmetric convex bodies. We develop two new ingredients: a pruning procedure and a flattening mechanism.