Manifolds of positive reach, differentiability, tangent variation, and attaining the reach

TL;DR

This paper provides an elementary proof linking positive reach of manifolds to local $C^{1,1}$ graph representations, improves bounds on tangent space variation, and generalizes reach attainment results beyond smooth manifolds.

Contribution

It offers a simpler proof of the relation between positive reach and $C^{1,1}$ graphs, refines Lipschitz bounds on tangent variation, and extends reach attainment results to broader sets.

Findings

Elementary proof of the positive reach and $C^{1,1}$ graph equivalence.

Optimal Lipschitz constants for tangent space angles.

Generalization of reach attainment from smooth to arbitrary positive reach sets.

Abstract

This paper contains three main results. Firstly, we give an elementary proof of the following statement: Let $M$ be a (closed, in both the geometrical and topological sense of the word) topological manifold embedded in $\mathbb{R}^d$. If $M$ has positive reach, then M can locally be written as the graph of a $C^{1,1}$ from the tangent space to the normal space. Conversely if $M$ can locally written as the graph of a $C^{1,1}$ function from the tangent space to the normal space, then $M$ has positive reach. The result was hinted at by Federer when he introduced the reach, and proved by Lytchak. Lytchak's proof relies heavily CAT(k)-theory. The proof presented here uses only basic results on homology. Secondly, we give optimal Lipschitz-constants for the derivative, in other words we give an optimal bound for the angle between tangent spaces in term of the distance between the points. This improves earlier results, that were either suboptimal or assumed that the manifold was $C^2$. Thirdly, we generalize a result by Aamari et al. which explains the how the reach is attained from the smooth setting to general sets of positive reach.