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The Voronoi functional is maximized by the Delaunay triangulation in the plane

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Abstract

We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.

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Author information

Authors and Affiliations

  1. IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria

    Herbert Edelsbrunner & Anton Nikitenko

  2. School of Mathematical & Statistical Science, The University of Texas Rio Grande Valley, Rio Grande Valley, Texas, USA

    Alexey Glazyrin & Oleg R. Musin

Authors
  1. Herbert Edelsbrunner
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  2. Alexey Glazyrin
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  3. Oleg R. Musin
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  4. Anton Nikitenko
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Corresponding author

Correspondence to Alexey Glazyrin.

Additional information

This research is partially supported by the Russian Government under the Mega Project 11.G34.31.0053, by the Toposys project FP7-ICT-318493-STREP, by ESF under the ACAT Research Network Programme, by RFBR grant 11-01-00735, and by NSF grants DMS-1101688, DMS-1400876.

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Edelsbrunner, H., Glazyrin, A., Musin, O.R. et al. The Voronoi functional is maximized by the Delaunay triangulation in the plane. Combinatorica 37, 887–910 (2017). https://doi.org/10.1007/s00493-016-3308-y

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  • DOI: https://doi.org/10.1007/s00493-016-3308-y

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