  A generalization of the cosine of the Friedrichs angle between two subspaces
to a parameter associated to several closed subspaces of a Hilbert space is
given. This parameter is used to analyze the rate of convergence in the von
Neumann-Halperin method of cyclic alternating projections. General dichotomy
theorems are proved, in the Hilbert or Banach space situation, providing
conditions under which the alternative QUC/ASC (quick uniform convergence
versus arbitrarily slow convergence) holds. Several meanings for ASC are
proposed.
