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Let $\mathcal{D}$ be a set of $n$ pairwise disjoint unit disks in the plane.We describe how to build a data structure for $\mathcal{D}$ so that for anypoint set $P$ containing exactly one point from each disk, we can quickly findthe onion decomposition (convex layers) of $P$.  Our data structure can be built in $O(n \log n)$ time and has linear size.Given $P$, we can find its onion decomposition in $O(n \log k)$ time, where $k$is the number of layers. We also provide a matching lower bound. Our solutionis based on a recursive space decomposition, combined with a fast algorithm tocompute the union of two disjoint onion

Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition