What conditions on the locations of cell centers (and the global boundary, if any) guarantee that the corresponding Voronoi cells are all different shapes?
When creating a jigsaw puzzle, we want distinct shapes so that the puzzle has a unique solution. Voronoi cells are nicely shaped. However, note that in general, distinct shapes do not guarantee a unique solution: for example, consider a 1xN rectangle composed of 1x(M_i) pieces, where the M's differ but sum to N.
Harder: an infinite sequence of points such that stopping the sequence at any point yields a Voronoi partition of cells of all different shapes.
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