  We prove a sharp dimension-free isoperimetric inequality involving the
Cheeger constant and the volume entropy $h$, in metric measure spaces with
non-negative Ricci curvature. For RCD spaces admitting a needle decomposition,
we show that the sharp isoperimetric inequality or the Cheeger constant is
achieved by a non-trivial measurable set, if and only if a one-dimensional
space with density $e^{h t}$ is split off. These results are proved in the
non-smooth framework using optimal transport techniques, but new even in the
smooth setting.
