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 DobbinsEmail author
  • 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 



B. Aronov supported by NSF Grants CCF-11-17336 and CCF-12-18791. O. Cheong and M. G. Dobbins supported by NRF Grant 2011-0030044 (SRC-GAIA) from the Government of Korea.


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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
    Email author
  • 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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