Timeline for Is the center of gravity in a CAT(0) space contained in the convex hull?
Current License: CC BY-SA 3.0
11 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 17, 2015 at 19:21 | history | edited | Anton Petrunin |
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| May 9, 2015 at 14:07 | vote | accept | Dylan Thurston | ||
| May 7, 2015 at 19:00 | comment | added | Benoît Kloeckner | @DylanThurston: my suggestion does need some more argument, but it seems to me that the CAT(0) condition should precisely imply that the objective function is as least as convex as in Euclidean space. | |
| May 7, 2015 at 16:24 | comment | added | Dylan Thurston | I like that suggestion, @BenoîtKloeckner, but it does need some more arguments. In particular, you would need to get quantitative about how strictly convex the objective function is. | |
| May 7, 2015 at 14:25 | comment | added | Benoît Kloeckner | @DylanThurston: I think that compactness arguments can be replaced by completeness in this kind of situation. I would try to use the argument for uniqueness to show that a minimizing sequence of points is Cauchy, and then concludes under a mere completeness assumption. It would need a little bit of checking though. | |
| May 7, 2015 at 13:54 | comment | added | Dylan Thurston | Now that I think a little more, are we guaranteed the existence of a center of gravity when $X$ is not locally compact? | |
| May 7, 2015 at 13:53 | answer | added | Dylan Thurston | timeline score: 11 | |
| May 7, 2015 at 13:09 | comment | added | Dylan Thurston | This is exactly right, thank you! That is Proposition II.2.4 of Bridson and Häfliger. (The notion of angle has to be suitably defined, of course.) | |
| May 7, 2015 at 12:36 | comment | added | user35593 | Just an idea: Assume that $x$ is not in the convex hull and let $x'=Px$ be the projection of $x$ onto the convex hull. Then we can try to show that $d(x',x_i)<d(x,x_i)$ for all $i$ to get a contradiction. Looking at the geodesic triangle $x_i,x,x'$ we can maybe prove that $\angle x_ix'x>90^\circ>\angle x'xx_i$ and follow that $d(x',x_i)<d(x,x_i)$. However I am not that familiar with CAT(0) spaces and can not rigorously prove this. | |
| May 7, 2015 at 9:18 | history | asked | Dylan Thurston | CC BY-SA 3.0 |