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Voronoi diagrams have interesting mathematical properties and applications in algorithms and modeling. But what are its applications in pure mathematics? For example, what theorems can be proved using Voronoi diagrams?

Both elementary and advanced applications are interesting for me.

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For topological combinatorics, Voronoi diagrams provide extremely nice configuration spaces. In particular, some mass partitioning problems can be tackled using this type of subdivision. Power diagrams (which extend Voronoi diagrams) are more commonly used for this purpose. See for instance the expository article of Günter Ziegler [1] where he explains his results from [2]. The problem of interest in those papers is the following conjecture of R. Ramana Rao

"Given any convex shape in the plane and any positive integer N. There exist some way(s) of partitioning this shape into N convex pieces so that they all have the same area and perimeter."

[1] Günter M. Ziegler. "Cannons at sparrows" Newsletter of the European Mathematical Society 95: 25-31

[2] Blagojević, Pavle VM, and Günter M. Ziegler. "Convex equipartitions via equivariant obstruction theory." Israel Journal of Mathematics 200.1 (2014): 49-77.

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    $\begingroup$ A proof of a similar result (also using power diagrams) is used in [3]. [3] Soberón, Pablo. "Balanced convex partitions of measures in R^d." Mathematika 58.01 (2012): 71-76. $\endgroup$
    – Pablo Soberón
    Commented May 9, 2015 at 13:46
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They are honestly used all over the place, it's easy for this to get out of control. Maybe one non-obvious place is the construction of the canonical triangulation of a hyperbolic 3-manifold:

Epstein, D. B. A., Penner, R. C., Euclidean decompositions of noncompact hyperbolic manifolds. J. Differential Geom. 27 (1988), no. 1, 67–80.

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  • $\begingroup$ To be honest, this is more about Delaunay triangulations/convex hulls than Voronoi diagrams. But of course they are intimately tied together. $\endgroup$
    – Dylan Thurston
    Commented May 9, 2015 at 14:25
  • $\begingroup$ Thanks. It's good if you give hints to help me find some applications. It will help me a lot. $\endgroup$
    – Ali Khezeli
    Commented May 9, 2015 at 17:44
  • $\begingroup$ More elementary ones are also good. $\endgroup$
    – Ali Khezeli
    Commented May 9, 2015 at 17:52

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