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The reach $\tau_M$ of a manifold $M$ is the largest number such that any point at distance less than $\tau_M$ from $M$ has a unique nearest point on $M$. This concept seems quite related to the local feature size introduced in computational geometry, but I haven't seen the two compared. Two questions:

Q1. Is the reach identical (under appropriate assumptions) to the local feature size?

Q2. What is the relationship (if any) between a positive reach $\tau > 0$ of a manifold $M$ and its $C^k$ smoothness? If $M$'s reach is $\tau > 0$, does that imply anything about $k$? If $M$ is $C^k$-differentiable, does that imply anything about $\tau$?

         
          Image from Hunting for Manifolds, Larry Wasserman.
I think the answer to Q1 is Yes, and to Q2 is None. But I await responses from those who are more familiar with these concepts.

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    $\begingroup$ Positive reach implies that the submanifold is $C^{1,1}$; that is, 1-st derivatives is Lipschitz. The Lipschitz constant fro derivative gives a positive bound on reach locally, but might fail to give right global bound if the submanifold wiggles and comes back close to it self. $\endgroup$
    – Anton Petrunin
    May 19, 2017 at 0:19
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    $\begingroup$ And in fact for hypersurfaces positive reach is equivalent to $C^{1,1}$, see arxiv.org/pdf/1304.4179. $\endgroup$
    – Igor Belegradek
    May 19, 2017 at 11:45
  • $\begingroup$ @IgorBelegradek; Thanks for the reference: Scholtes, Sebastian. "On hypersurfaces of positive reach, alternating Steiner formulae and Hadwiger's Problem." $\endgroup$
    – Joseph O'Rourke
    May 19, 2017 at 11:50
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    $\begingroup$ The differentiability is intrinsic property of manifold whereas its reach depends on its embedding. You can construct infinitely smooth embeded submanifolds of positive reach for any $\tau > 0$. Just take a unit circle and pull point $[-1,0]$ towards $[0,1]$. $\endgroup$
    – Vít Tuček
    May 19, 2017 at 12:17

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