Secondary Power Diagram, Dual of Secondary Polytope

  • Na Lei
  • Wei ChenEmail author
  • Zhongxuan Luo
  • Hang Si
  • Xianfeng Gu
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 131)


An ingenious construction of Gel’fand et al. (Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Basel, 1994) geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with non-empty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a non-degenerated coherent triangulation to another non-degenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are non-degenerated and coherent.


Upper envelope Convex hull Power diagram Weighted Delaunay triangulation Secondary polytope 


  1. 1.
    Zelevinsky, A.V., Gel’fand, I.M., Kapranov, M.M.: Newton polyhedra of principal a-determinants. Soviet Math. Dokl. 308, 20–23 (1989)MathSciNetGoogle Scholar
  2. 2.
    Billera, L.J., Filliman, P., Sturmfels, B.: Constructions and complexity of secondary polytopes. Adv. Math. 83(2), 155–179 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gel’fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Basel (1994)Google Scholar
  4. 4.
    Izmestiev, I., Klee, S., Novik, I.: Simplicial moves on balanced complexes. Adv. Math. 320, 82–114 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    D’Azevedo, E.F.: Optimal triangular mesh generation by coordinate transformation. SIAM J. Sci. Stat. Comput. 12(4), 755–786 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Joe, B.: Construction of three-dimensional Delaunay triangulations using local transformations. Comput. Aided Geom. Des. 8(2), 123–142 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Santos, F., De Loera, J.A., Rambau, J.: Triangulations, Structures for Algorithms and Applications. Springer, Berlin (2010)Google Scholar
  8. 8.
    Alexandrov, A.D.: Convex polyhedra. Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky. In: Springer Monographs in Mathematics. Springer, Berlin (2005)Google Scholar
  9. 9.
    Gu, X., Luo, F., Sun, J., Yau, S.-T.: Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge-Ampere equations. Asian J. Math. 20(2), 383–398 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Aurenhammer, F.: Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16(1), 78–96 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Na Lei
    • 1
  • Wei Chen
    • 2
    Email author
  • Zhongxuan Luo
    • 3
  • Hang Si
    • 4
  • Xianfeng Gu
    • 5
  1. 1.DUT-RU ISEDalian University of TechnologyDalianChina
  2. 2.School of Software TechnologyDalian University of TechnologyDalianChina
  3. 3.Key Laboratory for Ubiquitous Network and Service Software of Liaoning ProvinceDalianChina
  4. 4.Weierstrass Institute for Applied Analysis and Stochastics (WIAS)BerlinGermany
  5. 5.Department of Computer ScienceStony Brook UniversityStony BrookUSA

Personalised recommendations