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

Boris Aronov, Otfried Cheong, Michael Gene Dobbins +1

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 $\mathb…