The *reach* of a set $X\subseteq \mathbb{R}^d$ is the supremum of all $r \geq 0$ such that for all $y\in X^c$ with $dist(y,X)<r$ there is a unique $x\in X$ with $dist(y,x)= dist(y,X)$.

**My question:** If $X$ is an $m$-dimensional $C^{\infty}$ Riemannian manifold, is there an intrinsic definition of the reach, i.e., one that does not involve an ambient space?

**I know** that the reach is bounded from below by the inverse of the curvature. Furthermore, by Whitney embedding theorem, $X$ is embedded in $\mathbb{R}^{2m}$, so there is always an ambient space to be found. However, it is not clear whether all embeddings will have the same reach.

anyRiemannian manifold, $r$ can be made arbitrarily small. $\endgroup$ – Deane Yang Jan 9 at 15:58