Timeline for Optimal 8-vertex isoperimetric polyhedron?
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 12 at 21:11 | answer | added | vosramis | timeline score: 1 | |
| Sep 7 at 16:07 | answer | added | Oscar Lanzi | timeline score: 2 | |
| S Aug 20, 2017 at 4:07 | history | suggested | Martin Sleziak | CC BY-SA 3.0 |
added link to the paper
|
| Aug 20, 2017 at 2:49 | review | Suggested edits | |||
| S Aug 20, 2017 at 4:07 | |||||
| Aug 15, 2017 at 20:15 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image link broken; now fixed.
|
| Sep 6, 2014 at 2:14 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
replaced deprecated tag 'geometry'
|
| Sep 6, 2014 at 1:37 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Lost web image, now uploaded.
|
| Sep 8, 2011 at 19:10 | vote | accept | Joseph O'Rourke | ||
| Aug 30, 2011 at 4:22 | comment | added | Henry Cohn | @Agol: That would be great. I don't see how to write down a condition like this that isn't terribly complicated, but I may well be missing something clever. Incidentally, fixing the number of facets (rather than vertices) seems to lead to much cleaner results, so it might lead to a nicer analogy with CMC surfaces as well. For example, Lindelöf proved that the optimal polyhedron with a fixed number of facets is always circumscribed around a sphere. Then optimizing the isoperimetric ratio amounts to minimizing volume. | |
| Aug 30, 2011 at 0:57 | comment | added | Ian Agol | Is there any local restrictions on a minimizing polyhedron with fixed number of vertices known (analogous to a CMC surface being a local isoperimetric minimizing surface)? Presumably there is some local restrictions on the stars of vertices, so that if you wiggle the vertex in a volume preserving way, the area of the star increases. It would be nice if this condition was a discrete version of the CMC condition. | |
| Aug 29, 2011 at 23:52 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 15 characters in body
|
| Aug 29, 2011 at 23:47 | comment | added | Joseph O'Rourke | @Jean-Marc: Indeed! We all owe Marcel Berger a debt of gratitude for his "advertising" these elementary yet beautiful unsolved questions! He mentions, for example, how little is known for these isoperimetric problems in dimensions greater than 3. | |
| Aug 29, 2011 at 11:32 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 144 characters in body; added 4 characters in body
|
| Aug 29, 2011 at 10:49 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
edited body; added 65 characters in body; added 5 characters in body
|
| Aug 29, 2011 at 7:46 | comment | added | Jean-Marc Schlenker | @Joseph & Igor: thanks for the references! It striking how many elementary and beautiful questions are still open. | |
| Aug 29, 2011 at 5:14 | answer | added | Henry Cohn | timeline score: 18 | |
| Aug 28, 2011 at 23:49 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 37 characters in body
|
| Aug 28, 2011 at 23:23 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 102 characters in body
|
| Aug 28, 2011 at 23:15 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 365 characters in body
|
| Aug 28, 2011 at 23:06 | comment | added | Joseph O'Rourke | Turns out that for the special case of 8 vertices, the Mutoh reference (thanks, Igor!) only verifies the earlier solution of Berman and Haynes I cited above, which polyhedron has four vertices of degree 4 and four vertices of degree 5, and is not the cube, as is evident from the image I posted! | |
| Aug 28, 2011 at 17:19 | comment | added | Joseph O'Rourke | @Anton: I didn't mean to imply that existence is a known open question. Only that I didn't encounter a clear statement in my limited reading. | |
| Aug 28, 2011 at 17:02 | comment | added | Anton Petrunin | "it is known that there is an optimal polyhedron for each v". Is it stated as an open question somewhere? $$ $$ It is not a problem to show existence for algebraic volume (i.e., with overlaps counted). Further this optimal polyhedron (for large $v$) has to be close to a sphere. It should follow that this optimal polyhedron has no overlaps... | |
| Aug 28, 2011 at 16:04 | comment | added | Joseph O'Rourke | @Igor: Thanks! That reference includes a candidate, now included at the end of my question. | |
| Aug 28, 2011 at 15:50 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
added 314 characters in body; added 6 characters in body
|
| Aug 28, 2011 at 15:28 | comment | added | Igor Rivin | @Jean-Marc and @Joseph: more is known, see www-users.cs.umn.edu/~shao/fulltext.pdf | |
| Aug 28, 2011 at 15:20 | comment | added | Joseph O'Rourke | @Jean-Marc: It appears that question was settled by Berman and Haynes in "Volumes of polyhedra inscribed in the unit sphere in $\mathbb{R}^3$," Math. Ann. 1970, which I cannot access right now: springerlink.com/content/r7h7112424214257 | |
| Aug 28, 2011 at 15:01 | comment | added | Jean-Marc Schlenker | There is a related and also interesting question: what is the polyhedron with 8 vertices, inscribed in a sphere, of max volume? There is a candiate in www.jstor.org/stable/2003644 but I don't know whether the question has been solved since then. | |
| Aug 28, 2011 at 14:48 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |