A notion of robustness and stability of manifolds

Abstract

Starting from the notion of thickness of Parks we define a notion of robustness for arbitrary subsets of R-k and we investigate its relationship with the notion of positive reach of Federer. We prove that if a set M is robust, then its boundary a M is of positive reach and conversely (under very mild restrictions) if partial derivative M is of positive reach, then M is robust. We then prove that a closed non-empty robust set in R-k (different from R-k) is a codimension zero submanifold of class C-1 with boundary. As a partial converse we show that any compact codimension zero submanifold with boundary of class C-2 is robust. Using the notion of robustness we prove a kind of stability theorem for codimension zero compact submanifolds with boundary: two such submanifolds, whose boundaries are close enough (in the sense of Hausdorff distance), are diffeomorphic