  In this paper we study the $\Gamma$-limit, as $p\to 1$, of the functional
  $$
  J_{p}(u)=\frac{\displaystyle\int_\Omega |\nabla u|^p + \beta\int_{
  \partial \Omega} |u|^p}{\displaystyle \int_\Omega |u|^p},
  $$ where $\Omega$ is a smooth bounded open set in $\mathbb R^{N}$, $p>1$ and
$\beta$ is a real number. Among our results, for $\beta >-1$, we derive an
isoperimetric inequality for
  \[
  \Lambda(\Omega,\beta)=\inf_{u \in BV(\Omega), u\not \equiv 0}
\frac{\displaystyle |Du|(\Omega) + \min(\beta,1)\int_{
  \partial \Omega} |u|}{\displaystyle \int_\Omega |u|}
  \] which is the limit as $p\to 1^{+}$ of $ \lambda(\Omega,p,\beta)=
\displaystyle \min_{u\in W^{1,p}(\Omega)} J_{p}(u). $ We show that among all
bounded and smooth open sets with given volume, the ball maximizes
$\Lambda(\Omega, \beta)$ when $\beta \in$ $(-1,0)$ and minimizes
$\Lambda(\Omega, \beta)$ when $\beta \in[0, \infty)$.
