Invariance property of proximity conditions in nonlinear subdivision

Proximity conditions are used extensively in the analysis of smoothness and approximation order properties of subdivision schemes for manifold-valued data. While these properties under question are independent of choice of coordinates on the manifold, it is not known whether the proximity condition itself is invariant under arbitrary change of coordinates. In this note, we answer this question to the affirmative, i.e. we prove that the proximity condition is satisfied in one coordinate system if and only if it is satisfied in any other coordinate system. In passing, we prove a connection between the general proximity condition and an alternate proximity condition used in the interpolatory case. This interpolatory proximity condition also enjoys the same invariance under change of coordinates.