Let $M$ be a smooth closed $d$-manifold embedded in $\mathbb{R}^n$, and let $d_M: \mathbb{R}^n\to \mathbb{R}$ be given by $d_M(x) = \min_{m\in M} d(x,m)$.
Define $M\oplus \epsilon = \{ x\in \mathbb{R}^n \mid d_M(x) \leq \epsilon\}$. The reach of $M$, ${\rm Reach}(M)$, is the largest $\epsilon$ such that any element of $M\oplus \epsilon$ has a unique nearest point in $M$.
I have seen the assumption ${\rm Reach}(M) > 0$ quite a few places. Can someone provide me with a non-trivial example for which ${\rm Reach}(M) = 0$? Thanks!
The trick is being non-compact. For example $$M = \{(x,y) \in \mathbb{R} | xy = \pm 1\}$$ has reach 0 because $M$ approaches the $x$-axis which has no unique nearest point.