Is every tessellation of the plane into convex polygons a Voronoi tessellation?

Question

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

Answer 1

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.