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The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms

Author

Listed:
  • Supanut Chaidee

    (Advanced Research Center for Computational Simulation, Department of Mathematics, Faculty of Science, Chiang Mai University, 239 Huaykaew Road, Suthep District, Muang, Chiang Mai 50200, Thailand)

  • Kokichi Sugihara

    (Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan)

Abstract

Given a set of constrained vertex norms, we proved the existence of a convex configuration with respect to the set of distinct constrained vertex norms in the two-dimensional case when the constrained vertex norms are distinct or repeated for, at most, four points. However, we proved that there always exists a convex configuration in the three-dimensional case. In the application, we can imply the existence of the non-empty spherical Laguerre Voronoi diagram.

Suggested Citation

  • Supanut Chaidee & Kokichi Sugihara, 2020. "The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms," Mathematics, MDPI, vol. 8(4), pages 1-13, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:645-:d:349030
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    References listed on IDEAS

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    1. Dula, J. H. & Helgason, R. V., 1996. "A new procedure for identifying the frame of the convex hull of a finite collection of points in multidimensional space," European Journal of Operational Research, Elsevier, vol. 92(2), pages 352-367, July.
    2. Bahman Kalantari, 2015. "A characterization theorem and an algorithm for a convex hull problem," Annals of Operations Research, Springer, vol. 226(1), pages 301-349, March.
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