  We study the method of cyclic projections when applied to closed and linear
subspaces $M_i$, $i=1,\ldots,m$, of a real Hilbert space $\mathcal H$. We show
that the average distance to individual sets enjoys a polynomial behaviour
$o(k^{-1/2})$ along the trajectory of the generated iterates. Surprisingly,
when the starting points are chosen from the subspace
$\sum_{i=1}^{m}M_i^\perp$, our result yields a polynomial rate of convergence
$\mathcal O(k^{-1/2})$ for the method of cyclic projections itself. Moreover,
if $\sum_{i=1}^{m} M_i^\perp$ is not closed, then both of the aforementioned
rates are best possible in the sense that the corresponding polynomial
$k^{1/2}$ cannot be replaced by $k^{1/2+\varepsilon}$ for any $\varepsilon >0$.
