While the original question Convex triangulations was aimed at the existence and calculation of convex triangulations of a given set of $n$ points in the Euclidean plane, I would like to ask the opposite

Question:

where do pointsets, that come close to the ideal of having a convex Delaunay triangulation, "naturally" occur?

It appears to me that the pointsets that are generated via Weighted Voronoi Stippling algorithms diagrams or the nuclei of 2D biological cell structures are almost free of non-convex maximal unions of Delaunay triangles with a common inner points as a vertex.