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

@article{Aronov2017TheNO, title={The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions}, author={B. Aronov and O. Cheong and M. G. Dobbins and X. Goaoc}, journal={Discrete & Computational Geometry}, year={2017}, volume={57}, pages={104-124} }

We show that the union of n translates of a convex body in $$\mathbb {R}^3$$R3 can have $$\varTheta (n^3)$$Θ(n3) holes in the worst case, where a hole in a set X is a connected component of $$\mathbb {R}^3 \setminus X$$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.

2 Citations

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

- Computer Science, Mathematics
- 2016

2

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