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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)

Abstract

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.

Keywords

Upper envelope Convex hull Power diagram Weighted Delaunay triangulation Secondary polytope 

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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

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