Timeline for One-step problems in geometry
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 30, 2010 at 19:38 | comment | added | Anton Petrunin | Petya, this statement which you can do is everything known about the problem (if I remember right my conversation with Sergei Tabachnikov). | |
| Jan 1, 2010 at 18:14 | comment | added | Petya | I can prove your problem for a case when l (or r) is a constant function. Another variant (generalization) of the initial problem, unknown to me, is the following. Consider (locally) affine coordinate x on the inner curve, which is an angle of a tangent line with a fixed line. This coordinate is defined uniquely up to a sum with constant. Then, for any angle a an equation l(x)=r(x+a) is solvable. | |
| Jan 1, 2010 at 18:01 | comment | added | Petya | I did not know that problem and could not solve it (even if outer curve is convex). I can comment it: Each point x of inner curve defines "left" and "right" segment of a chord tangent at x and corresponding values l(x) and r(x). Consider images L and R of functions l and r correspondingly. If L \subset R or R \subset L then for any diffeomorphism g of the inner curve there exists x such that L(g(x))=R(x). It seems that, in general, it is not true that L \subset R either R \subset L. | |
| Dec 29, 2009 at 4:33 | comment | added | Anton Petrunin | Not bad --- I solved it :). BTW do you know that a similar problem is open: find a point on outer curve which has two tangent segments from this point to the inner curve has equal size (at least if outer curve is not convex). | |
| Dec 29, 2009 at 3:33 | history | answered | Petya | CC BY-SA 2.5 |