Metric Spaces of Shapes and Geometries Constructed from Set Parametrized Functions

In modelling, optimization, control, or identification problems with respect to a family of subsets of a fixed hold-all in RN, the nice vector space structure of the calculus of variations and control theory is no longer available. The family of all subsets of the Euclidean space is a group for the algebraic symmetric difference. One way to construct a family of variable domains is to consider the images of a fixed subset of RN by some family of transformations of RN. The group structure for the composition of transformations induces a group structure on the space of variable sets. Each variable set is parametrized by its associated transformation (homeomorphism or diffeomorphism). So it is topologically identical to the initial set. To get around this limitation, metric spaces of set-parametrized functions (characteristic, distance, oriented distance, support function) have been introduced (Hausdorff metric associated with the distance function and Caccioppoli sets with the characteristic function) but many more complete metric spaces can be constructed such as, for instance, the sets of positive reach or the sets of bounded curvatures. The paper surveys past and current constructions while introducing new metrics for families of sets or of embedded submanifolds with a prescribed smoothness yet allowing topological changes.