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On the Structure of Higher Order Voronoi Cells

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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.

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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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Authors and Affiliations

  1. Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona, and BGSMath, Barcelona, Spain

    Juan Enrique Martínez-Legaz

  2. School of Mathematics and Statistics, UNSW Sydney, Sydney, Australia

    Vera Roshchina

  3. Department of Physics and Mathematics, UDLAP, Puebla, Mexico

    Maxim Todorov

  4. Institute of Mathematics and Informatics, BAS, Sofia, Bulgaria

    Maxim Todorov

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  1. Juan Enrique Martínez-Legaz
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  2. Vera Roshchina
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  3. Maxim Todorov
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Correspondence to Vera Roshchina.

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Martínez-Legaz, J.E., Roshchina, V. & Todorov, M. On the Structure of Higher Order Voronoi Cells. J Optim Theory Appl 183, 24–49 (2019). https://doi.org/10.1007/s10957-019-01555-2

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  • DOI: https://doi.org/10.1007/s10957-019-01555-2

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