Skip to main content
SpringerLink
Log in

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

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

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.

References

  1. Agarwal, P.K., Pach, J., Sharir, M.: State of the union (of geometric objects). In: Proceedings of the Joint Summer Research Conference on Discrete and Computational Geometry: 20 Years Later. Contemporary Mathematics, vol. 452, pp. 9–48. AMS, Providence, RI (2008)

  2. Agarwal, P.K., Har-Peled, S., Kaplan, H., Sharir, M.: Union of random Minkowski sums and network vulnerability analysis. Discrete Comput. Geom. 52(3), 551–582 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aronov, B., Sharir, M.: On translational motion planning of a convex polyhedron in 3-space. SIAM J. Comput. 26(6), 1785–1803 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aronov, B., Cheong, O., Dobbins, M.G., Goaoc, X.: The number of holes in the union of translates of a convex set in three dimensions. In: 32nd International Symposium on Computational Geometry (SoCG 2016), Leibniz International Proceedings in Informatics (LIPIcs), vol. 51, pp. 10:1–10:16 (2016)

  5. Aurenhammer, F.: Voronoi diagrams: a survey of a fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991)

    Article  Google Scholar 

  6. Borsuk, K.: On the imbedding of systems of compacta in simplicial complexes. Fundam. Math. 35, 217–234 (1948)

    MathSciNet  MATH  Google Scholar 

  7. Clarkson, K.L., Varadarajan, K.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43–58 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Efrat, A., Sharir, M.: On the complexity of the union of fat convex objects in the plane. Discrete Comput. Geom. 23(2), 171–189 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, Chapter 23, 2nd edn, pp. 513–528. CRC Press, Boca Raton (2004)

    Google Scholar 

  10. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  11. Icking, C., Klein, R., Lé, N.M., Ma, L.: Convex distance functions in 3-space are different. Fundam. Inform. 22, 331–352 (1995)

    MathSciNet  MATH  Google Scholar 

  12. Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal ostacles. Discrete Comput. Geom. 1(1), 59–71 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Koltun, V., Sharir, M.: Polyhedral Voronoi diagrams of polyhedra in three dimensions. Discrete Comput. Geom. 31, 83–124 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kovalev, M.D.: Svoistvo vypuklykh mnozhestv i ego prilozhenie (A property of convex sets and its application). Mat. Zametki 44, 89–99 (1988)

    MathSciNet  Google Scholar 

  15. Mitchell, J.S.B., O’Rourke, J.: Computational geometry column. Int. J. Comput. Geom. Appl. 11(05), 573–582 (2001)

    Article  MathSciNet  Google Scholar 

  16. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Menlo Park (1984)

    MATH  Google Scholar 

  17. Pach, J., Tardos, G.: On the boundary complexity of the union of fat triangles. SIAM J. Comput. 31(6), 1745–1760 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sharir, M.: Algorithmic motion planning. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, Chapter 47, 2nd edn, pp. 1037–1064. CRC Press, Boca Raton (2004)

    Google Scholar 

  19. Sharir, M., Agarwal, P.K.: Davenport–Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (2010)

    MATH  Google Scholar 

Download references

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.

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Otfried Cheong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Gene Dobbins
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xavier Goaoc
    View author publications

    You can also search for this author in PubMed Google Scholar

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].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

Search

Navigation