Submanifolds of class $C^{1,\alpha}$ and sets with positive $\mu$-reach

TL;DR

This paper extends the concept of positive reach from smooth submanifolds to less regular ones using $$-reach, providing quantitative bounds on the gradient of the distance function for $C^{1,lpha}$ submanifolds.

Contribution

It demonstrates that $C^{1,lpha}$ submanifolds have positive $$-reach and establishes sharp quantitative estimates on the gradient of the distance function.

Findings

$C^{1,lpha}$ submanifolds have positive $$-reach.

Quantitative bounds on the gradient of the distance function are derived.

The exponent in the bounds is shown to be sharp.

Abstract

It is well-known since the seminal work of Herbert Federer [Trans. of the AMS, 1959] that submanifolds of class $C^{1,1}$ have positive reach. In this paper, we extend this property to less regular submanifolds by using the notion of $\mu$-reach that was introduced in the 2000's. We first show that every compact $C^1$ submanifold of the Euclidean space $\E^n$ has positive $\mu$-reach for all $\mu<1$. We then show that intermediate regularities $C^{1,\alpha}$ induce more quantitative results on the norm $\|\nabla \d_M\|$ of the generalized gradient of the distance function~$\d_M$ to the submanifold. More precisely, if $M\subset \E^n$ is a submanifold of class $C^{1,\alpha}$, with $\alpha<1$, then there exists a constant $C>0$ such that $$\forall p\in\E^n\setminus M,\quad 1 - \| \nabla \d_M(p) \|^2 \leq C ~ \d_M(p)^{\frac{2 \alpha}{1- \alpha}}.$$ We finally show that the exponent $2\alpha/(1-\alpha)$ in this estimate is sharp.