Timeline for Hidden convexity
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 26, 2012 at 0:52 | vote | accept | Igor Rivin | ||
| Jan 31, 2011 at 20:40 | comment | added | Bill Thurston | @Igor: Deane's answer is good, sorry for being lazy. A general technique for finding critical points with $k$ negative eigenvalues to the 2nd derivative is to look at $k$-dimensional submanifolds, and let them flow downhill by the geodesic flow. The sometimes catch onto index k critical points, and spread out along the unstable manifold. For the example in the picture, which is improper, there are straight lines with more than one tangency to the level sets, so the function restricted to the straight line has more than one critical point so is not convex. | |
| Jan 31, 2011 at 16:46 | comment | added | Deane Yang | Igor, for your first question, the idea is to look at all paths joining the two local minima and minimize among all the paths the maximum value of the function along each path. You find a saddle point this way. | |
| Jan 31, 2011 at 16:40 | comment | added | Igor Rivin | @BBill: thanks much, this is very enlightening. A couple of questions: in the first part, what do you mean by a minimax path? (the argument is geometrically fairly clear, but...) In the "improper" example, what is the invariant that tells you that the function is not conjugate to a convex function? I am not quite seeing it... | |
| Jan 31, 2011 at 14:13 | comment | added | Deane Yang | Bill, thanks for the clarifications! | |
| Jan 31, 2011 at 4:57 | history | edited | Bill Thurston | CC BY-SA 2.5 |
Addressed revised question: opposite answer if proper is also required.
|
| Jan 31, 2011 at 4:43 | comment | added | Bill Thurston | @Igor: I actially wondered whether you meant $\Omega \rightarrow \Omega_1$, and I probably should have worded it conditionally on the intended meaning. I suspect you also intended (or should have intended) to make $f$ a proper function --- if so, that would get rid of the second example, and change the game. | |
| Jan 31, 2011 at 4:15 | comment | added | Igor Rivin | @Bill: my apologies, I misspoke (see Another Edit in the statement) -- it has been a while since I thought about this. I think that makes your peanut example not work; I have not quite absorbed your second example... | |
| Jan 31, 2011 at 2:13 | comment | added | Bill Thurston | @Deane Yang. Sorry. I'm thinking of $f$ transported by $\phi$, so the formula is $f(\phi^{-1}(x))$; I suppose it's a misnomer to call this "conjugation", although it would be nice to have a single term applying in general to the concept of transforming a structure by a homeomorphism. | |
| Jan 31, 2011 at 1:48 | comment | added | Deane Yang | Quick question: What do you mean by "conjugate"? | |
| Jan 31, 2011 at 1:43 | history | answered | Bill Thurston | CC BY-SA 2.5 |