The finest regular coarsening and recursively-regular subdivisions

We generalize the notion of regular polyhedral subdivision of a point (or vector) configuration in a new direction. This is done after studying some related objects, like the finest regular coarsening and the regularity tree of a subdivision. Properties of these two objects are derived, which confer more structure to the class of non-regular subdivisions, relating them to its (in a sense) closest regular subdivision. We introduce the class of recursively-regular subdivisions and show that it is a proper superclass of the regular subdivisions and a proper subclass of the visibility-acyclic subdivisions. We also show that recursivelyregular triangulations of a given configuration are in general not connected by geometric bistellar flips. Finally, some algorithms related to these new concepts are given and applications of the main results of the article are pointed out. In particular, relations to covering by floodlights, covering by homotheties, tensegrity of spider webs and a high-dimensional graph embedding problem are presented.