Q:

Do Voronoi cells "converge" towards their centroid? 

If you do the following:

Place random points on a finite surface
Draw the Voronoi diagram using the points as germs
Calculate the centroid of each cell
Repeat step 2 and 3 indefinitely using the centroids obtained in step 3 as germs. 

How does that look like?  Does it converge to a diagram where all the germs are in the centroid of their cell?  I suspect there's a convergence most of the times, but is there a set of starting points where it doesn't converge? 
Secondary question: Does the fact that the surface is finite change anything?  I suspect it does, considering borders affects the position of the centroid around the edge (so after a few iterations, it affects all cells) but not the Voronoi cell. 

A:

This paper answers the question (and goes far beyond). 
Reformulating theorem 2.1 tells you that, for every finite set of random points over a bounded surface, the iterated centroids will converges to some distribution.  Although, as the paper mentions, the distribution limit may not be optimal in the sense that they do not necessarily split the surface evenly (see quantization error in the paper). 
Unboundedness implies a lot of theoretical complications, which is in fact the main point of interest of this paper.  Convergence can still be ensured by adding constraints to your initial set of points.