  We obtain new sharp isoperimetric inequalities on a Riemannian manifold
equipped with a probability measure, whose generalized Ricci curvature is
bounded from below (possibly negatively), and generalized dimension and
diameter of the convex support are bounded from above (possibly infinitely).
Our inequalities are sharp for sets of any given measure and with respect to
all parameters (curvature, dimension and diameter). Moreover, for each choice
of parameters, we identify the model spaces which are extremal for the
isoperimetric problem. In particular, we recover the Gromov--L\'evy and
Bakry--Ledoux isoperimetric inequalities, which state that whenever the
curvature is strictly positively bounded from below, these model spaces are the
$n$-sphere and Gauss space, corresponding to generalized dimension being $n$
and $\infty$, respectively. In all other cases, which seem new even for the
classical Riemannian-volume measure, it turns out that there is no single model
space to compare to, and that a simultaneous comparison to a natural one
parameter family of model spaces is required, nevertheless yielding a sharp
result.
