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 R-3 can have Theta (n(3)) holes in the worst case, where a hole in a set X is a connected component of R-3 \ 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.