  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 a
connected component of $\mathbb{R}^3 \setminus X$. This refutes a 20-year-old
conjecture. As a consequence, we also obtain improved lower bounds on the
complexity of motion planning problems and of Voronoi diagrams with convex
distance functions.
