  Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the
problem of minimizing the ratio between the $s-$perimeter and the
$N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the
nonlocal version of the well-known Cheeger problem. We prove various properties
of optimal sets for this problem, as well as some equivalent formulations. In
addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue
problems is investigated, in relation with this optimization problem. The
presentation is as self-contained as possible.
