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

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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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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Mathematics Subject Classification (2000)

  • 52C20
  • 52C22