Timeline for Reach of manifold vs. $C^k$-manifold
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| May 19, 2017 at 12:17 | comment | added | Vít Tuček | 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 $[0,1]$. | |
| May 19, 2017 at 11:50 | comment | added | Joseph O'Rourke | @IgorBelegradek; Thanks for the reference: Scholtes, Sebastian. "On hypersurfaces of positive reach, alternating Steiner formulae and Hadwiger's Problem." | |
| May 19, 2017 at 11:45 | comment | added | Igor Belegradek | And in fact for hypersurfaces positive reach is equivalent to $C^{1,1}$, see arxiv.org/pdf/1304.4179. | |
| May 19, 2017 at 0:19 | comment | added | Anton Petrunin | 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. | |
| May 18, 2017 at 23:11 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |