  In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson
coordinates in [Gillette et al., AiCM, doi:10.1007/s10444-011-9218-z], we prove
interpolation error estimates for the mean value coordinates on convex polygons
suitable for standard finite element analysis. Our analysis is based on
providing a uniform bound on the gradient of the mean value functions for all
convex polygons of diameter one satisfying certain simple geometric
restrictions. This work makes rigorous an observed practical advantage of the
mean value coordinates: unlike Wachspress coordinates, the gradients of the
mean value coordinates do not become large as interior angles of the polygon
approach pi.
