Timeline for The bounded complex of a polyhedral decomposition
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 9, 2023 at 13:49 | vote | accept | Nicholas Proudfoot | ||
| Mar 7, 2023 at 21:10 | comment | added | Nicholas Proudfoot | @JohnWiltshire-Gordon By "polyhedron" I mean an intersection of finitely many closed half-spaces. In particular, it is convex. | |
| Mar 7, 2023 at 20:13 | comment | added | John Wiltshire-Gordon | @NicholasProudfoot Ok, so I'm not allowed to subdivide faces. Are non-convex polyhedra allowed? I could point the parallel rays away from each other and get an hourglass-type separation between the vertices. | |
| Mar 7, 2023 at 19:56 | comment | added | Nicholas Proudfoot | @JohnWiltshire-Gordon That would be no good because some of the rays would be contained in the strip, but would not be faces of the strip. In particular, condition 3 would fail. | |
| Mar 7, 2023 at 19:29 | comment | added | John Wiltshire-Gordon | @NicholasProudfoot Well then I will draw a capacitor diagram consisting of two vertices each with three outgoing rays; an infinite strip separates the two vertices. | |
| Mar 7, 2023 at 19:25 | comment | added | Sam Hopkins | I believe Pavel Galashin is the person who once showed me the argument using discrete Morse theory. You could possibly try contacting him. | |
| Mar 7, 2023 at 19:24 | comment | added | Nicholas Proudfoot | @JohnWiltshire-Gordon Touché. I guess I would like to assume that my decomposition contains at least one vertex. | |
| Mar 7, 2023 at 19:23 | comment | added | Nicholas Proudfoot | @SamHopkins Thanks for the suggestion! I have thought about that a little bit, but was deterred by the fact that discrete Morse theory is usually applied to simplicial complexes rather than to polyhedral complexes. | |
| Mar 7, 2023 at 19:19 | comment | added | John Wiltshire-Gordon | What are the rules for polyhedra? Is V a polyhedral decomposition of itself into one part? | |
| Mar 7, 2023 at 15:09 | comment | added | Sam Hopkins | If I recall correctly, there is a relatively straightforward proof for that exercise from Stanley using discrete Morse theory. Perhaps discrete Morse theory can also be applied to the situation you are interested in. | |
| Mar 7, 2023 at 15:07 | answer | added | M. Winter | timeline score: 1 | |
| Mar 6, 2023 at 18:27 | comment | added | Nicholas Proudfoot | Thanks to Dane Miyata for pointing out that the statement in Stanley's notes is proved in Theorem Theorem 4.5.7(i) of the book Oriented Matroids. Still, I'd like to know what happens when $\mathscr{P}$ does not come from a hyperplane arrangement! | |
| Mar 6, 2023 at 18:08 | history | asked | Nicholas Proudfoot | CC BY-SA 4.0 |