  We carry on an analysis of the size of the contact surface of a Cheeger set
$E$ with the boundary of its ambient space $\Omega$. We show that this size is
strongly related to the regularity of $\partial \Omega$ by providing bounds on
the Hausdorff dimension of $\partial E\cap \partial\Omega$. In particular we
show that, if $\partial \Omega$ has $C^{1,\alpha}$ regularity then
$\mathcal{H}^{d-2+\alpha}(\partial E\cap \partial\Omega)>0$. This shows that a
sufficient condition to ensure that $\mathcal{H}^{d-1}(\partial E\cap \partial
\Omega)>0$ is that $\partial \Omega$ has $C^{1,1}$ regularity. Since the
Hausdorff bounds can be inferred in dependence of the regularity of $\partial
E$ as well, we obtain that $\Omega$ convex, which yields $\partial E\in
C^{1,1}$, is also a sufficient condition. Finally, we construct examples
showing that such bounds are optimal in dimension $d=2$.
