Submanifolds of class $C^1$ and sets with positive mu-reach

Abstract: The geometry of smooth submanifolds of the Euclidean space of dimension $n$ is well known. However, a consequence of the famous Nash-Kuiper theorem established in 1954-55 implies that there exists infinitely many submanifolds that have a $C^1$ regularity and that cannot be of class $C^2$. The goal of this talk is to investigate the study of $C^1$ submanifolds using the geometric notion of mu-reach that was introduced in the 2000's in the field of geometric inference. 

The notion of mu-reach is a generalization of the notion of reach that was introduced by Herbert Federer in 1959 in order to generalize the notion of curvature measures to non-smooth and non convex objects. Federer mentioned that submanifolds of class $C^{1,1}$ have positive reach. In this talk, we will start with an introduction on the notions of reach and mu-reach. We will then show that submanifolds of class $C^1$ have the property of having positive mu-reach.