On a New Proximity Condition for Manifold-Valued Subdivision Schemes

An open theoretical problem in the study of subdivision algorithms for approximation of manifold-valued data has been to give necessary and sufficient conditions for a manifold-valued subdivision scheme, based on a linear subdivision scheme, to share the same regularity as the linear scheme. This is called the smoothness equivalence problem. In a companion paper, the authors introduced a differential proximity condition that solves the smoothness equivalence problem. In this paper, we review this condition, comment on a few of its unanticipated features, and as an application, show that the single basepoint log-exp scheme suffers from an intricate breakdown of smoothness equivalence. We also show that the differential proximity condition is coordinate independent, even when the linear scheme is not assumed to possess the relevant smoothness.