Dimensional lower bounds for contact surfaces of Cheeger sets

We carry out an analysis of the size of the contact surface between a Cheeger set E and its ambient space Ω⊂Rd. By providing bounds on the Hausdorff dimension of the contact surface ∂E∩∂Ω, we show a fruitful interplay between this size itself and the regularity of the boundaries. Eventually, we obtain sufficient conditions to infer that the contact surface has positive (d−1) dimensional Hausdorff measure. Finally we prove by explicit examples in two dimensions that such bounds are optimal.