Timeline for Shortcutting quasigeodesics
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 14, 2021 at 10:38 | vote | accept | Ashot Minasyan | ||
| Oct 14, 2021 at 8:48 | answer | added | Florian Lehner | timeline score: 4 | |
| Oct 12, 2021 at 21:14 | comment | added | Ashot Minasyan | @FlorianLehner: you are right, thanks for your comments. If you decide to put them as an answer I would be happy to accept it. | |
| Oct 12, 2021 at 15:26 | comment | added | Florian Lehner | @AshotMinasyan: If I'm not mistaken, then a path is a $(1,c)$-quasigeodesic if and only if its length exceeds the distance between its endpoints by at most $c$, i.e. we only need to check the whole path, not all sub-paths. This property is certainly preserved under replacing a subpath by a shorter (or equally long) one. | |
| Oct 12, 2021 at 14:43 | comment | added | Ashot Minasyan | @FlorianLehner: thanks, that's a great counter-example for both questions! I have thought about the same path, but treated it as a $(1,2k)$-quasigeodesic, instead of changing the multiplicative constant! What if I require $\lambda=1$? | |
| Oct 12, 2021 at 14:31 | comment | added | Florian Lehner | here is an easy counterexample in $\mathbb Z^2$: the path [$k$ steps up, $k$ steps right, $k$ steps down] is a $(3,0)$ quasi-geodesic, but replacing the first $2k$ steps by the geodesic [$(k-1)$ steps right, $k$ steps up, one step right] does not preserve this. | |
| Oct 12, 2021 at 14:20 | comment | added | markvs | How about ${\mathbb Z}^2$? There are many geodesics with the same endpoints there. | |
| Oct 12, 2021 at 14:08 | history | asked | Ashot Minasyan | CC BY-SA 4.0 |