  If $\Omega \subset \R^n$ is a smooth bounded domain and $q \in (0,
\frac{n}{n-1})$ we consider the Poincare-Sobolev inequality \[ c
\Bigl(\int_{\Omega} \abs{u}^\frac{n}{n-1}\Bigr)^{1-\frac{1}{n}} \le
\int_{\Omega} \abs{Du}, \] for every $u \in \mathrm{BV}(\Omega)$ such that
$\int_{\Omega} \abs{u}^{q-1} u = 0$. We show that the sharp constant is
achieved. We also consider the same inequality on an $n$--dimensional compact
Riemannian manifold $M$. When $n \ge 3$ and the scalar curvature is positive at
some point, then the sharp constant is achieved. In the case $n \ge 2$, we need
the maximal scalar curvature to satisfy some strict inequality.
