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

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 $$\forall p\in\E^n\setminus M,\quad 1 - \| \nabla \d_M(p) \|^2 \leq C ~ \d_M(p)^{\frac{2 α}{1- α}}.$$ We finally show that the exponent $2\alpha /(1-\alpha )$ in this estimate is sharp.