  In view of the great performance of circumcentered isometry methods for
solving the best approximation problem, in this work we further investigate the
locally proper circumcenter mapping and circumcentered method. Various examples
of locally proper circumcenter mapping are presented and studied. Inspired by
some results on circumcentered-reflection method by Behling, Bello-Cruz, and
Santos in their recent papers, we provide sufficient conditions for one step
convergence of circumcentered isometry methods for finding the best
approximation point onto the intersection of fixed point sets of related
isometries. In addition, we elaborate the performance of circumcentered
reflection methods induced by reflectors associated with hyperplanes and
halfspaces for finding the best approximation point onto (or a point in) the
intersection of hyperplanes and halfspaces.
