Dimensional lower bounds for contact surfaces 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 $\mathrm{\Omega }$. We show that this size is strongly related to the regularity of $\mathrm{\partial }\mathrm{\Omega }$ by providing bounds on the Hausdorff dimension of $\mathrm{\partial }E\cap \mathrm{\partial }\mathrm{\Omega }$. In particular we show that, if $\mathrm{\partial }\mathrm{\Omega }$ has $C^{1,\alpha }$ regularity then ${\mathcal{H}}^{d-2+\alpha }(\mathrm{\partial }E\cap \mathrm{\partial }\mathrm{\Omega })>0$. This shows that a sufficient condition to ensure that $$\mathcal{H}$^{d-1}(\partial E\cap \partial \Omega)>0$isthat$\partial \Omega$has$C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in dependence of the regularity of $\partial E$aswell,weobtainthat$\Omega$convex,whichyields$\partial E\in C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.