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

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 ${\text{E}}^n$ has positive $\mu$-reach for all $\mu <1$. We then show that intermediate regularities $C^{1,\alpha }$ induce more quantitative results on the norm $\mathrm{\parallel }\mathrm{\nabla }{\text{d}}_M\mathrm{\parallel }$ of the generalized gradient of the distance function~${\text{d}}_M$ to the submanifold. More precisely, if $M\subset {\text{E}}^n$ is a submanifold of class $C^{1,\alpha }$, with $\alpha <1$, then there exists a constant $C>0$ such that $$\mathrm{\forall }p\in {\text{E}}^n\setminus M,1-\mathrm{\parallel }\mathrm{\nabla }{\text{d}}_M(p){\mathrm{\parallel }}^2\le C{\text{d}}_M(p{)}^{\frac{2\alpha }{1-\alpha }}\mathrm{.}$$ We finally show that the exponent $2\alpha \mathrm{/}(1-\alpha )$ in this estimate is sharp.