Discrete & Computational Geometry

, Volume 57, Issue 1, pp 104–124 | Cite as

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

  • Boris Aronov
  • Otfried Cheong
  • Michael Gene Dobbins
  • Xavier Goaoc


We show that the union of n translates of a convex body in \(\mathbb {R}^3\) can have \(\varTheta (n^3)\) 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.


Union complexity Convex sets Motion planning 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Boris Aronov
    • 1
  • Otfried Cheong
    • 2
  • Michael Gene Dobbins
    • 3
  • Xavier Goaoc
    • 4
  1. 1.Department of Computer Science and Engineering, Tandon School of EngineeringNew York UniversityBrooklynUSA
  2. 2.KAISTDaejeonSouth Korea
  3. 3.Department of Mathematical SciencesBinghamton UniversityBinghamtonUSA
  4. 4.Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE, UPEMMarne-la-ValléeFrance

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