  The elementary Euclidean concept of circumcenter has recently been employed
to improve two aspects of the classical Douglas--Rachford method for projecting
onto the intersection of affine subspaces. The so-called
circumcentered-reflection method is able to both accelerate the average
reflection scheme by the Douglas--Rachford method and cope with the
intersection of more than two affine subspaces. We now introduce the technique
of circumcentering in blocks, which, more than just an option over the basic
algorithm of circumcenters, turns out to be an elegant manner of generalizing
the method of alternating projections. Linear convergence for this novel
block-wise circumcenter framework is derived and illustrated numerically.
Furthermore, we prove that the original circumcentered-reflection method
essentially finds the best approximation solution in one single step if the
given affine subspaces are hyperplanes.
