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

TL;DR

This paper proves that the union of n translates of a convex set in three dimensions can have on the order of n^3 holes, refuting a long-standing conjecture and impacting related computational geometry problems.

Contribution

It establishes the maximum number of holes in unions of convex translates in 3D, providing a counterexample to a 20-year-old conjecture and improving bounds in motion planning and Voronoi diagrams.

Findings

Union of n convex translates can have Θ(n^3) holes in 3D.

Refutes a 20-year-old conjecture on the complexity of such unions.

Improves lower bounds on motion planning and Voronoi diagram complexity.

Abstract

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.