  We characterize the sets $\XX$ of all products $PQ$, and $\YY$ of all
products $PQP$, where $P,Q$ run over all orthogonal projections and we solve
the problems $\arg\min\{\|P-Q\|: (P,Q) \in \cal Z\}$, for $\cal Z=\XX$ or
$\YY.$ We also determine the polar decompositions and Moore-Penrose
pseudoinverses of elements of $\XX.$
