Subsets of Positive Reach

In previous chapters, we have seen that it is possible to define curvatures which describe the global shape of two classes of subsets of E^N, namely the convex bodies and the smooth submanifolds. A good challenge is to find larger classes of subsets on which a more general theory holds. In 1958, Federer [43] made a major advance in two directions:

• 1. He could define a large class of subsets on which it is possible to define curvatures, extending the class of smooth submanifolds and convex bodies. The subsets of E^N belonging to this class are called ^{subsets with positive reach}. Basically, the main observation is that the important tool in this context is the orthogonal projection onto the studied subset. For a given convex body, this orthogonal projection is defined at every point of E^N and, for a smooth submanifold, it is defined on a neighborhood of it. Federer defined the class of subsets which admit locally this property, even if they are neither convex nor smooth, calling them subsets of positive reach.

• 2. He realized that this new theory could be considered as a particular (signed) measure theory on E^N