  This work presents several new results concerning the analysis of the
convergence of binary, univariate, and linear subdivision schemes, all related
to the {\it contractivity factor} of a convergent scheme. First, we prove that
a convergent scheme cannot have a contractivity factor lower than half. Since
the lower this factor is, the faster is the convergence of the scheme, schemes
with contractivity factor $\frac{1}{2}$, such as those generating spline
functions, have optimal convergence rate.
  Additionally, we provide further insights and conditions for the convergence
of linear schemes and demonstrate their applicability in an improved algorithm
for determining the convergence of such subdivision schemes.
