Hausdorff compactness and regularity for classes of open sets under geometric constraints

This article introduces innovative classes of open sets in ${\mathbb{R}}^N$, where $N=2,3$, characterized by a geometric property associated with the inward normal. The focus lies on proving compactness results for the Hausdorff topology within these classes. Furthermore, the paper establishes the equivalence of convergences, encompassing Hausdorff, compact, and characteristic functions, for select classes. We also investigate the regularity of the thickness function associated with these domains and analyze how the regularity of the fixed convex set $C$ influences the boundary regularity of the admissible shapes.