  Kolmogorov $n$-widths and low-rank approximations are studied for families of
elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay
of the $n$-widths can be controlled by that of the error achieved by best
$n$-term approximations using polynomials in the parametric variable. However,
we prove that in certain relevant instances where the diffusion coefficients
are piecewise constant over a partition of the physical domain, the $n$-widths
exhibit significantly faster decay. This, in turn, yields a theoretical
justification of the fast convergence of reduced basis or POD methods when
treating such parametric PDEs. Our results are confirmed by numerical
experiments, which also reveal the influence of the partition geometry on the
decay of the $n$-widths.
