  Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with
smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat
conformal metric. We prove that the supremum of the isoperimetric quotient over
the scalar-flat conformal class is strictly larger than the best constant of
the isoperimetric inequality in the Euclidean space, and consequently is
achieved, if either (i) $n\ge 12$ and $\partial M$ has a nonumbilic point; or
(ii) $n\ge 10$, $\partial M$ is umbilic and the Weyl tensor does not vanish at
some boundary point.
