Shape quality for generalized barycentric interpolation

At a time when generalized barycentric coordinates are gaining popularity in an ever-growing set of application contexts, there is a need for a clear understanding of the relationship between the interpolation properties of various coordinate types and the geometries on which they are defined. Consider, for instance, an academic or industrial code employing generalized barycentric coordinates that fails to give a numerical output when applied to some polygonal or polyhedral mesh. Could the premature termination of the simulation be averted by changing the coordinates that are used? Or by improving the “quality” of the mesh elements in some way? Or neither?.