I think(?) that this paper addresses your question, if you stipulate
a constant density distribution:

> Balzer, Michael, Thomas Schlömer, and Oliver Deussen. Capacity-constrained point distributions: A variant of Lloyd's method. Vol. 28. No. 3. *ACM SIGGRAPH*, 2009.
([ACM link](http://dl.acm.org/citation.cfm?id=1531392)).

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[![Fig3][1]][1]
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> "[...] This constraint enforces that each point in a distribution has the same capacity. Intuitively, the capacity can be understood as the area of the point’s corresponding Voronoi region weighted with the given density function. By demanding that each point’s capacity is the same, we ensure that each point obtains equal importance in the resulting distribution."

I believe this is a form of an *optimal transport problem*.


  [1]: https://i.sstatic.net/Mw9H9.jpg