  We consider the problem of determining the manifold $n$-widths of Sobolev and
Besov spaces with error measured in the $L_p$-norm. The manifold widths control
how efficiently these spaces can be approximated by general non-linear
parametric methods with the restriction that the parameter selection and
parameterization maps must be continuous. Existing upper and lower bounds only
match when the Sobolev or Besov smoothness index $q$ satisfies $q\leq p$ or $1
\leq p \leq 2$. We close this gap and obtain sharp lower bounds for all $1 \leq
p,q \leq \infty$ for which a compact embedding holds. A key part of our
analysis is to determine the exact value of the manifold widths of finite
dimensional $\ell^M_q$-balls in the $\ell_p$-norm when $p\leq q$. Although this
result is not new, we provide a new proof and apply it to lower bounding the
manifold widths of Sobolev and Besov spaces. Our results show that the
Bernstein widths, which are typically used to lower bound the manifold widths,
decay asymptotically faster than the manifold widths in many cases.
