The number of holes in the union of translates of a convex set in three dimensions

Abstract

We show that the union of n translates of a convex body in ℝ3 can have Θ(n3) holes in the worst case, where a hole in a set X is a connected component of ℝ3 \ 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.