Timeline for Characterization of convexity by connectedness of hyperplane sections
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 29 at 14:03 | vote | accept | Dmitrii Korshunov | ||
| Jan 29 at 13:58 | comment | added | Dmitrii Korshunov | this is funny how it's actually true for $n=1$ in light of the characterization of closed subsets mathoverflow.net/questions/28967/… because an (at least reasonable) open set in $\mathbb R^2$ is contractible iff its boundary is connected | |
| Jan 29 at 13:54 | comment | added | Dmitrii Korshunov | thank you, this is cool! just a little elaboration for those (like me) who didn't see it right away: 1) if a hyperplane section is disconnected, then the sections by all nearby hyperplanes are also disconnected -- hence it is enough to prove for Morse linear functions. 2) once we pass a critical value of index less than dimension of the levelset ($\le 2n-2$ in our case) the contentedness doesn't change, e.g. because $H_{d-1}(\partial M^d)$ is not affected by gluing in a $d-2$-dimensional cell. | |
| Jan 29 at 1:27 | history | answered | Anton Petrunin | CC BY-SA 4.0 |