Shape Analysis via Oriented Distance Functions

The object of this paper is twofold. We first present constructions which induce topologies on subsets of a fixed domain or hold-all D in RN by using set parameterized functions in an appropriate function space. Second, we study the role of the family of oriented distance functions (also known as algebraic or signed distance functions) in the analysis of shape optimization problems. They play an important role in the introduction of topologies which retain the classical geometric properties associated with sets: convexity, exterior normals, mean curvature, Ck boundaries, etc.