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Given an arbitrary tessellation of the two-dim. plane into convex polygons, can one show that this is always a Voronoi tessalation, without knowing the points that would define the Voronoi tessellation?

If this is not the case, what are the sufficient conditions that a given tessellation into convex polygons is a Voronoi tessellation?

Thanks

Greg Martin
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TomS
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  • If you start with the infinite square grid with vertices at $\Bbb Z^2$, and then, for every $a,b\in \Bbb Z$, you replace the four squares with a vertex at $(4a,4b)$ with a single square of side-length $2$, the resulting tesselation is not Voronoi. – Greg Martin Jul 09 '20 at 08:01
  • No, tessellations of convex polygons are not always Voronoi tessellations. If you want an example, consider a large regular polygon surrounded by very small-diameter polygons. Points in the regular polygon near the boundary will be very close to some of the small polygons. This makes the placement of a point for the large regular polygon impossible. I can't answer your other question though. – user804886 Jul 09 '20 at 08:11
  • Thanks a lot. Can anybody answer the second question re sufficient conditions? except for the trivial one that „there exist points for which the Voronoi-condition holds“. – TomS Jul 09 '20 at 10:11

1 Answers1

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There aren't simple, easy to state sufficient conditions. But this question has been studied pretty thoroughly in this paper:

Ash, Peter F.; Bolker, Ethan D., Recognizing Dirichlet tesselations, Geom. Dedicata 19, 175-206 (1985). ZBL0572.52022.

The authors summarize this as:

To decide whether a given tessellation is a Dirichlet tessellation requires two steps: a local argument at each vertex and a pasting mechanism.

Given a tessellation, the generally describe a locally conclude that a tessellation is not Voronoi (or Dirichlet in their terminology). But things become especially complex to handle cases where the Voronoi sites cannot be determined from the tessellation.

Alex
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