Abstract
Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain “dominance” map. Asano, Matoušek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.
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References
Asano, T., Kirkpatrick, D.: Distance trisector curves in regular convex distance metrics. In: Proceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering, pp. 8–17. IEEE Computer Society, California (2006)
Asano T., Matoušek J., Tokuyama T.: Zone diagrams: existence, uniqueness, and algorithmic challenge. SIAM J. Comput. 37(4), 1182–1198 (2007)
Asano T., Matoušek J., Tokuyama T.: The distance trisector curve. Adv. Math. 212(1), 338–360 (2007)
Aurenhammer F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)
Benyamini Y., Lindenstrauss J.: Nonlinear Functional Analysis, vol. I, Colloquium Publications 48. American Mathematical Society (AMS), Providence (2000)
de Biasi, S.C., Kalantari, B., Kalantari, I.: Maximal zone diagrams and their computation. In: Proceedings of the 7th International Symposium on Voronoi Diagrams in Science and Engineering, pp. 171–180. IEEE Computer Society, California (2010)
Chun J., Okada Y., Tokuyama T.: Distance trisector of a segment and a point. Interdiscip. Inf. Sci. 16(1), 119–125 (2010)
Kadets M.I., Levitan B.M.: Banach space. In: Hazewinkel, M. (eds) Encyclopedia of Mathematics. Springer, Berlin (2002). http://eom.springer.de/b/b015190.htm
Kawamura, A., Matoušek, J., Tokuyama, T.: Zone diagrams in Euclidean spaces and in other normed spaces. arXiv:0912.3016v1 (Preprint, 2009)
Kopecká, E., Reem, D., Reich, S.: Zone diagrams in compact subsets of uniformly convex normed spaces. Israel J. Math. (2011). doi:10.1007/s11856-011-0094-5
Okabe A., Boots B., Sugihara K., Chiu S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Probability and Statistics, 2nd edn. Wiley, New York (2000)
Reem D., Reich S.: Zone and double zone diagrams in abstract spaces. Colloquium Mathematicum 115(1), 129–145 (2009)
Tarski A.: A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5, 285–309 (1955)
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Kawamura, A., Matoušek, J. & Tokuyama, T. Zone diagrams in Euclidean spaces and in other normed spaces. Math. Ann. 354, 1201–1221 (2012). https://doi.org/10.1007/s00208-011-0761-1
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DOI: https://doi.org/10.1007/s00208-011-0761-1