Suppose $(X^m, g)$ is a closed Riemannian manifold of dimension $m$ with the following properties,
There is a constant $\kappa$ such that $\kappa r^m \leq Vol(B(x, r)) \leq \kappa^{-1} r^m$ for every $r \in (0,1)$.
For every $C^1$ function $f$, we have $(\int_X |f|^{\frac{2m}{m-2}})^{\frac{m-2}{m}} \leq C_S (\int_X |f|^2 + |\nabla f|^2)$.
For every $C^1$ function $f$, we have $\int_X |f|^2 \leq C_P (\int_X |\nabla f|^2 + (\int_X f)^2)$.
Can we say that there is a positive constant $I$ depending only on $m, \kappa, C_S, C_P$ such that $\frac{Area(\partial \Omega)^m}{Vol(\Omega)^{m-1}} \geq I$ for every domain $\Omega$ satisfying $Vol(\Omega) \leq \frac{1}{2} Vol(X)$?
