Submanifolds of class $C^{1,α}$ and sets with positive $μ$-reach

TL;DR

The work extends Federer's classical reach theory to the μ-reach framework, enabling the study of less regular submanifolds. It proves that compact C^1 submanifolds possess positive μ-reach for any μ<1 and provides quantitative bounds on the generalized gradient of the distance function for intermediate regularities, with a sharp exponent depending on α. By connecting μ-reach to offsets and curvature measures, the paper bridges geometric inference stability with intrinsic regularity, and it demonstrates the sharpness of the obtained exponent through explicit curve constructions. These results broaden the applicability of curvature-measure ideas to a wider class of geometric objects and offer precise control over distance-function regularity near C^{1,α} submanifolds.

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 $μ$-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 $μ$-reach for all $μ<1$. We then show that intermediate regularities $C^{1,α}$ 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 $C^{1,α}$, with $α<1$, 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 $2α/(1-α)$ in this estimate is sharp.

Figures (7)

• Figure 1: Generalized gradient of distance function $\mathrm{d}_K$.
• Figure 2: Left: the parabola $x^2$ and its medial axis (blue). Right: graph of its critical function. Since $\mathrm{reach}(K) = \frac{1}{2}$ we have $\chi(d)=1$ for $d\leqslant \frac{1}{2}$.
• Figure 3: Example of a non compact $C^1$ curve whose $\mu$-reach vanishes for $\mu > \frac{1}{2}$.
• Figure 4: Proof of Theorem \ref{['theo: C1_implique_continuite_fonction_crit']}.
• Figure 5: The construction of the proof.

Theorems & Definitions (56)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Proposition 2.1
• Definition 2.3
• Proposition 2.2
• Remark 2.1
• Definition 2.4