Products of orthogonal projections and polar decompositions

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