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

Discrete & Computational Geometry volume 57, pages 104–124 (2017)Cite this article

Abstract

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.

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Notes

  1. We say a path \(\pi \) is convex in a direction u if the orthogonal projection of \(\pi \) onto \(u^\bot \) is injective and the set \(\pi + \mathbb {R}^+ u\) is convex.

  2. In other words, the i-dimensional simplices of \(\Delta \) form a basis of the vector space \({{\mathrm{\mathcal {C}}}}_i(\Delta )\), which consists of formal sums of i-simplices, each simplex being assigned a real coefficient.

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Acknowledgments

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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Authors and Affiliations

  1. Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA

    Boris Aronov

  2. KAIST, Daejeon, South Korea

    Otfried Cheong

  3. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, USA

    Michael Gene Dobbins

  4. Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE, UPEM, 77454, Marne-la-Vallée, France

    Xavier Goaoc

Authors
  1. Boris Aronov
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  2. Otfried Cheong
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  3. Michael Gene Dobbins
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  4. Xavier Goaoc
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Corresponding author

Correspondence to Michael Gene Dobbins.

Additional information

Editor in Charge: János Pach

An extended abstract of this work was presented at the 32nd International Symposium on Computational Geometry [4].

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Aronov, B., Cheong, O., Dobbins, M.G. et al. The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions. Discrete Comput Geom 57, 104–124 (2017). https://doi.org/10.1007/s00454-016-9820-4

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  • DOI: https://doi.org/10.1007/s00454-016-9820-4

Keywords

  • Union complexity
  • Convex sets
  • Motion planning