Israel Journal of Mathematics

, Volume 188, Issue 1, pp 1–23

Zone diagrams in compact subsets of uniformly convex normed spaces

Article
First Online:
Received:
Revised:

Abstract

A zone diagram is a relatively new concept which has emerged in computational geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a certain mapping, and neither its uniqueness nor its existence are obvious in advance. It has been studied by several authors, starting with T. Asano, J. Matoušek and T. Tokuyama, who considered the Euclidean plane with singleton sites, and proved the existence and uniqueness of zone diagrams there. In the present paper we prove the existence of zone diagrams with respect to finitely many pairwise disjoint compact sites contained in a compact and convex subset of a uniformly convex normed space, provided that either the sites or the convex subset satisfy a certain mild condition. The proof is based on the Schauder fixed point theorem, the Curtis-Schori theorem regarding the Hilbert cube, and on recent results concerning the characterization of Voronoi cells as a collection of line segments and their geometric stability with respect to small changes of the corresponding sites. Along the way we obtain the continuity of the Dom mapping as well as interesting and apparently new properties of Voronoi cells.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. Asano and D. Kirkpatrick, Distance trisector curves in regular convex distance metrics, Proceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2006), pp. 8–17.Google Scholar
  2. [2]
    T. Asano, J. Matoušek and T. Tokuyama, The distance trisector curve, Advances in Mathematics 212 (2007), 338–360, a preliminary version in STOC 2006, pp. 336–343.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    T. Asano, J. Matoušek and T. Tokuyama, Zone diagrams: Existence, uniqueness, and algorithmic challenge, SIAM Journal on Computing 37 (2007), 1182–1198, a preliminary version in SODA 2007, pp. 756–765.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    T. Asano and T. Tokuyama, Drawing equally-spaced curves between two points, Abstracts of the 14th Annual Workshop on Computational Geometry with a Focus on Open Problems, 2004, pp. 24–25.Google Scholar
  5. [5]
    F. Aurenhammer, Voronoi diagrams-a survey of a fundamental geometric data structure, ACM Computing Surveys, Vol. 3, 1991, pp. 345–405.CrossRefGoogle Scholar
  6. [6]
    S. C. de Biasi, B. Kalantari and I. Kalantari, Maximal zone diagrams and their computation, Proceedings of the 7th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2010), pp. 171–180.Google Scholar
  7. [7]
    J. Chun, Y. Okada and T. Tokuyama, Distance trisector of segments and zone diagram of segments in a plane, Proceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007), pp. 66–73.Google Scholar
  8. [8]
    J. A. Clarkson, Uniformly convex spaces, Transactions of the American Mathematical Society 40 (1936), 396–414.MathSciNetCrossRefGoogle Scholar
  9. [9]
    D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fundamenta Mathematicae 101 (1978), 19–38.MathSciNetMATHGoogle Scholar
  10. [10]
    J. Dugundji and A. Granas, Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.MATHGoogle Scholar
  11. [11]
    K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and AppliedMathematics, Vol. 83, Marcel Dekker Inc., New York, 1984.Google Scholar
  12. [12]
  13. [13]
    A. Illanes and S. B. Nadler Jr, Hyperspaces. Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 216, Marcel Dekker Inc., New York, 1999.Google Scholar
  14. [14]
    K. Imai, A. Kawamura, J. Matoušek, Y. Muramatsu and T. Tokuyama, Distance ksectors and zone diagrams, Extended abstract in EuroCG 2009, pp. 191–194. Available at http://www.cs.toronto.edu/?kawamura/publ/090317.
  15. [15]
    A. Kawamura, J. Matoušek and T. Tokuyama, Zone diagrams in Euclidean spaces and in other normed spaces, Mathematische Annalen, to appear. arXiv 0912.3016 (2009), a short version in SoCG 2010, pp. 216–221.Google Scholar
  16. [16]
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II: Function Spaces, Springer, Berlin, 1979.MATHGoogle Scholar
  17. [17]
    J. Munkres, Topology, second ed., Prentice Hall, Upper Saddle River, NJ, 2000.MATHGoogle Scholar
  18. [18]
    A. Okabe, B. Boots, K. Sugihara and S. N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, second ed., Wiley Series in Probability and Statistics, John Wiley & Sons Ltd., Chichester, 2000, with a foreword by D. G. Kendall.MATHGoogle Scholar
  19. [19]
    D. Reem, An algorithm for computing Voronoi diagrams of general generators in general normed spaces, Proceedings of the 6th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2009), pp. 144–152.Google Scholar
  20. [20]
    D. Reem, Voronoi and zone diagrams, Ph.D. thesis, The Technion-Israel Institute of Technology, 2010.Google Scholar
  21. [21]
    D. Reem, the geometric stability of Voronoi diagrams with respect to small changes of the sites, arXiv 1103.4125.Google Scholar
  22. [22]
    D. Reem and S. Reich, Zone and double zone diagrams in abstract spaces, Colloquium Mathematicum 115 (2009), 129–145, arXiv 0708.2668 (2007).MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Mathematica 2 (1930), 171–180.MATHGoogle Scholar
  24. [24]
    T. Zamfirescu, Typical starshaped sets, Aequationes Mathematicae 36 (1988), 188–200.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Institute of MathematicsCzech Academy of SciencesPragueCzech Republic
  2. 2.Institut für AnalysisJohannes Kepler UniversitätLinzAustria
  3. 3.Department of MathematicsThe Technion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations