  By using optimal mass transport theory we prove a sharp isoperimetric
inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic
volume growth at infinity. Our result extends recently proven isoperimetric
inequalities for normed spaces and Riemannian manifolds to a nonsmooth
framework. In the case of $n$-dimensional Riemannian manifolds with nonnegative
Ricci curvature, we outline an alternative proof of the rigidity result of S.
Brendle (2021). As applications of the isoperimetric inequality, we establish
Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in
Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate
symmetrization techniques and optimal volume non-collapsing properties. The
equality cases in the latter inequalities are also characterized by stating
that sufficiently smooth, nonzero extremal functions exist if and only if the
Riemannian manifold is isometric to the Euclidean space.
