Timeline for Convex curves with many inscribed triangles maximizing perimeter
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2020 at 15:57 | comment | added | Richard Montgomery | @Robert: (and the world): there is a 2006 paper by Baryshikov and Zharnitsky (MR2250493) which uses this same link (excuse the pun) between billiards and non-integrable distributions towards a beautiful end, proving results on building tables with a given caustic. | |
| Dec 6, 2012 at 16:28 | vote | accept | Pietro Majer | ||
| Oct 19, 2011 at 16:23 | comment | added | Michael Bächtold | @Robert: thanks for the comment and the references. I have to think more about what you wrote. | |
| Oct 19, 2011 at 15:37 | comment | added | Robert Bryant | @Joseph: Thanks. The curves don't have to be smooth (but I like them better that way). I guess you could, in principle, construct ones that are merely $C^1$ by using $C^1$-but-not-$C^2$ constraints to generate the solutions, but I don't know whether I could rig it to be pieces of ellipses. Maybe it would be worth writing a short note about this problem, because it is a somewhat interesting application of a technique that appears not to be well-known. | |
| Oct 19, 2011 at 15:32 | comment | added | Robert Bryant | @Michael: If you weren't careful, yes, they could split, into, say, 3 circles. However, I am working on the quotient space $T$ of triangles (actually, it turns out to be easier to work with the oriented triangles, a $\mathbb{Z}_3$ quotient), not the space of 'marked' triangles, which is an open subset of three copies of the plane. When you work on this space, this splitting problem doesn't arise for deformations. 'Regular' is in the usual sense of nonholonomic systems (which this is). Bliss is a good reference, but you can also consult Bryant-Hsu, Inventiones M. 114 (1993), 435–461. | |
| Oct 19, 2011 at 14:45 | comment | added | Michael Bächtold | Absolutely beautiful! Though one thing I don't see yet: a closed integral curve in $T$ could in principle give rise to 3 different closed curves in the plane, one for each vertex. Are there examples for this? Could a perturbation of an ellipse split up like this? Plus I'd be grateful if you can add a reference (or definition) of "regular" integral curve. | |
| Oct 19, 2011 at 11:00 | comment | added | Joseph O'Rourke | This is an impressive analysis! It would make a nice paper. I assume these curves $C$ are smooth? Or, for example, could they be composed of pieces of ellipses joined together? | |
| Oct 18, 2011 at 23:19 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Completed the sketch of the answer
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| Oct 15, 2011 at 16:54 | history | answered | Robert Bryant | CC BY-SA 3.0 |