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}
}
  • B. Aronov, O. Cheong, +1 author X. Goaoc
  • Published 2017
  • Computer Science, Mathematics
  • Discrete & Computational Geometry
  • 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. 

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