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


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    A. V. Akopyan: Extremal properties of Delaunay triangulations, Trudy ISA RAS 46 (2009), 174–187.

    Google Scholar 

  2. [2]

    R. Chen, Y. Xu, C. Gotsman and L. Liu: A spectral characterization of the Delaunay triangulation, Comput. Aided Geom. Design 27 (2010), 295–300.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    N. P. Dolbilin, H. Edelsbrunner, A. Glazyrin and O. R. Musin: Functionals on triangulations of Delaunay sets, Moscow Math. J. 14 (2014), 491–504.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    H. Edelsbrunner: Geometry and Topology for Mesh Generation, Cambridge Univ. Press, Cambridge, England, 2001.

    Book  MATH  Google Scholar 

  5. [5]

    T. Lambert: The Delaunay triangulation maximizes the mean inradius, in: Proc. 6th Canad. Conf. Comput. Geom., 1994, 201–206.

    Google Scholar 

  6. [6]

    C. L. Lawson: Software for C1 surface interpolation, in: Mathematical Software III, ed.: J. R. Rice, Academic Press, New York, 1977, 161–194.

    Google Scholar 

  7. [7]

    O. R. Musin: Properties of the Delaunay triangulation, in: Proc. 13th Ann. Sympos. Comput. Geom., 1997, 424–426.

    Google Scholar 

  8. [8]

    O. R. Musin: About optimality of Delaunay triangulations. In Geometry, Topology, Algebra and Number Theory, Applications, Conf. dedicated to 120th anniversary of B. N. Delone, (2010), 166–167.

    Google Scholar 

  9. [9]

    V. V. Prasolov: Problems in Plane Geometry–M.: MCCME, 2006.

    Google Scholar 

  10. [10]

    V. T. Rajan: Optimality of the Delaunay triangulation in Rd, Discrete Comput. Geom. 12 (1994), 189–202.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    S. Rippa: Minimal roughness property of the Delaunay triangulation, Comput. Aided Geom. Design 7 (1990), 489–497.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    R. Sibson: Locally equiangular triangulations, Comput. J. 21 (1978), 243–245.

    MathSciNet  Article  Google Scholar 

Download references

Author information



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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

Mathematics Subject Classification (2000)

  • 52C20
  • 52C22