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

TL;DR

This paper introduces new classes of open sets in \\mathbb{R}^N with geometric properties, proving compactness in Hausdorff topology and analyzing boundary regularity influenced by convex set properties.

Contribution

It establishes compactness results and convergence equivalences for these classes, linking geometric constraints to shape regularity and boundary behavior.

Findings

Proves Hausdorff compactness for the introduced classes.

Shows equivalence of Hausdorff, compact, and characteristic function convergences.

Analyzes how convex set regularity affects boundary regularity.

Abstract

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.