Products of orthogonal projections and polar decompositions

We characterize the sets X of all products PQ, and Y of all products PQP, where P,Q run over all orthogonal projections and we solve the problems argmin{‖P-Q‖:(P,Q)∈Z}, for Z=X or Y. We also determine the polar decompositions and Moore–Penrose pseudoinverses of elements of X.