Timeline for Secondary polytope
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 16, 2022 at 21:52 | vote | accept | André Henriques | ||
| Oct 12, 2022 at 22:17 | answer | added | Francisco Santos | timeline score: 4 | |
| Jun 5, 2022 at 21:00 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| May 6, 2022 at 21:00 | comment | added | Sam Hopkins | You are right that it does not make much sense to talk about integer points for the secondary polytope in general, but note that for the bigger ("universal") polytope that projects to the secondary polytope, its vertices are integer points by definition (and actually it only depends on the oriented matroid structure of the original point set, not the specific geometry, unlike the secondary polytope). | |
| May 6, 2022 at 20:30 | answer | added | Sam Hopkins | timeline score: 1 | |
| May 6, 2022 at 15:14 | comment | added | André Henriques | @SamHopkins. I'm not sure what you mean by an integer point of the secondary polytope... Were you assuming that $P$ is itself a polytope whose points have integer coordinates (I certainly wasn't assuming that)? What about the case when $P$ has rational coordinates? Can you then also say something along the lines of what you said in your second comment – a correspondence between arbitrary rational points and some kind triangulations?... | |
| May 6, 2022 at 15:08 | comment | added | André Henriques | My mistake for forgetting the adjective "regular". I just added it to the wording of my question. | |
| May 6, 2022 at 14:44 | history | edited | André Henriques | CC BY-SA 4.0 |
added 8 characters in body
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| May 6, 2022 at 13:11 | comment | added | Sam Hopkins | Okay, I think the paper "The Polytope of All Triangulations of a Point Configuration" by de Loera, Hosten, Santos, and Sturmfels (math.uni-bielefeld.de/documenta/vol-01/04.pdf) may explain a bit more precisely what I said above: there is a (much bigger) polytope whose points correspond to all triangulations of $P$, and the secondary polytope is a projection of this bigger polytope, so integer points of the secondary polytope can in general be associated to triangulations, but not in a bijective fashion. | |
| May 6, 2022 at 12:59 | comment | added | Sam Hopkins | Vertices correspond to regular triangulations. I believe there is a sense in which other lattice points correspond to (non-regular) triangulations, but the correspondence is not exact (i.e., not one-to-one). | |
| May 6, 2022 at 12:57 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise title
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| May 6, 2022 at 12:51 | history | asked | André Henriques | CC BY-SA 4.0 |