Convex combination maps over triangulations, tilings, and tetrahedral meshes

In a recent paper by the first author, a simple proof was given of a result by Tutte on the validity of barycentric mappings, recast in terms of the injectivity of piecewise linear mappings over triangulations. In this note, we make a short extension to the proof to deal with arbitrary tilings. We also give a simple counterexample to show that convex combination mappings over tetrahedral meshes are not necessarily one-to-one. AMS subject classification: Primary: 05C10, 05C85, Secondary: 65D17, 58E20.