The wellknown [Delaunay Triangulation](https://en.wikipedia.org/wiki/Delaunay_triangulation) $DT$ has as a straight line dual the also wellknown [Voronoi Diagram](https://en.wikipedia.org/wiki/Voronoi_diagram) $VD$.  

Both are most commonly defined in the Euclidean plane and are primarily beneficial for solving problems related to geometric proximity, but they have also been generalized to higher dimensions, different distance measures and also for problems related to $n$-th nearest neighbor queries up to farthest neighbor queries.   

This question is however only related to the planar euclidean case.
<br>
>**Question:**  

>Have analogous "tandems" of triangulations and convex polygonal neighborhoods of discrete pointsets in the euclidean plane already appeared, resp., investigated where the corners of the polygonal neighborhoods are not the circum centers of triangles (which can also be outside of triangles e.g. in random triangulations), but other special points associated to triangles, like the ones listed in the [ETC](http://faculty.evansville.edu/ck6/encyclopedia/ETC.html) (of which *the incircle center seems the most interesting choice* because it is guaranteed to be inside the most degenerated triangle with nonzero area and, because it seems to generate polygonal neighborhoods with longer short sides)?

I am especially interested in results related to optimality criteria that the selection of specific triangle points (apart from the circum center) implies for the generated polygonal diagrams.