Necessary conditions for the convergence of subdivision schemes with finite masks

Knowing that the convergence of the subdivision scheme with a nonnegative mask relies on the location of its support of the mask, we consider the positions of the points in the support and the convex cover of the support. We demonstrate the different properties between the inner and boundary points of the support for the mask, when the corresponding subdivision scheme converges. Furthermore, we find out that the so-called connectivity of a matrix A deduced by a given mask is some simple condition to guarantee those properties for nonnegative masks.