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We show that the union of $n$ translates of a convex body in $\mathbb{R}^3$can have $\Theta(n^3)$ holes in the worst case, where a hole in a set $X$ is aconnected component of $\mathbb{R}^3 \setminus X$. This refutes a 20-year-oldconjecture. As a consequence, we also obtain improved lower bounds on thecomplexity of motion planning problems and of Voronoi diagrams with convexdistance functions.

The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions