Contact surface of Cheeger sets

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 $partial Ecap partialOmega$. In particular we show that, if $partial Omega$ has $C^{1,alpha}$ regularity then $mathcal{H}^{d-2+alpha}(partial Ecap partialOmega)>0$. This shows that a sufficient condition to ensure that $mathcal{H}^{d-1}(partial Ecap partial Omega)>0$ 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 $partial Ein C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.