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On the Structure of Higher Order Voronoi Cells

  • Juan Enrique Martínez-Legaz
  • Vera RoshchinaEmail author
  • Maxim Todorov
Article
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Abstract

The classic Voronoi cells can be generalized to a higher order version by considering the cells of points for which a given k-element subset of the set of sites consists of the k closest sites. We study the structure of the k-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher order Voronoi cells for four points.

Keywords

Higher order Voronoi cells Structure of Voronoi cells 

Notes

Acknowledgements

We are grateful to the two referees for their thoughtful and thorough corrections and suggestions, including the proof of Proposition 3.10. These corrections have greatly improved the quality of our paper. The first author was supported by the MINECO of Spain, Grant MTM2014-59179-C2-2-P, and the Severo Ochoa Programme for Centres of Excellence in R&D [SEV-2015-0563]. He is affiliated to MOVE (Markets, Organizations and Votes in Economics). He thanks The School of Mathematics and Statistics of UNSW Sydney for sponsoring a visit to Sidney to complete this work. The second author is grateful to the Australian Research Council for continuous financial support via grants DE150100240 and DP180100602, which in particular sponsored a trip to Barcelona that initiated this collaboration. The third author was partially supported by MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P, and Sistema Nacional de Investigadores, Mexico.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departament d’Economia i d’Història EconòmicaUniversitat Autònoma de Barcelona, and BGSMathBarcelonaSpain
  2. 2.School of Mathematics and StatisticsUNSW SydneySydneyAustralia
  3. 3.Department of Physics and MathematicsUDLAPPueblaMexico
  4. 4.Institute of Mathematics and Informatics, BASSofiaBulgaria

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