Timeline for Infinite convex combinations in a Banach space
Current License: CC BY-SA 4.0
15 events
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| Sep 15, 2022 at 17:00 | history | edited | Glorfindel | CC BY-SA 4.0 |
broken link fixed, cf. https://meta.mathoverflow.net/q/5301/70594
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| Jan 25, 2016 at 7:11 | answer | added | Mikhail Ostrovskii | timeline score: 1 | |
| Jan 26, 2014 at 23:14 | history | edited | user9072 |
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| Jan 26, 2014 at 22:38 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Feb 22, 2011 at 14:55 | comment | added | Bill Johnson | I should mention that the $\sigma$-convex hull of a set came up naturally in the classification of Banach spaces that have the Radon-Nikodym property through Maynard's concept of $s$-dentability. Later Davis and Phelps showed that you could use just dentability, so these days no one talks about $s$-dentability. Their paper jstor.org/pss/2040618 gives the history, including the work of Rieffel that started this line of investigation. Or see the book Vector Measures by Diestel and Uhl. | |
| Feb 22, 2011 at 12:17 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| Feb 22, 2011 at 12:06 | comment | added | Pietro Majer | @Bill: actually your example suggests a simple counterexample to my naive conjecture (bounded, convex, and Baire => $\sigma$-convex), in form of a convex subset $C$, $B\subset C\subset \bar B$ between an open, non-strictly convex ball, and its closure. Such a set $C$ is certainly Baire, but it may fail to be $\sigma$-convex, if one face of its is not. | |
| Feb 22, 2011 at 11:37 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| Feb 22, 2011 at 7:29 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| Feb 22, 2011 at 7:28 | comment | added | Pietro Majer | Good point. I've added a remark, just to explain why I become interested in the subject. | |
| Feb 22, 2011 at 7:21 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| Feb 21, 2011 at 22:22 | comment | added | Bill Johnson | Notice that any subset of the closed unit ball of a Hilbert space (or any strictly convex Banach space) that contains the open unit ball is $\sigma$-absolutely convex, so $\sigma$-absolutely convex sets can be complicated topologically. | |
| Feb 21, 2011 at 22:16 | comment | added | Bill Johnson | @Andrew: You are describing the $\sigma$-absolutely convex sets. @Pietro Majer: Continuing what Andrew started, IMO the $\sigma$-convex sets are best described as the image of the positive part of the unit ball of an $\ell_1$ space. | |
| Feb 21, 2011 at 10:50 | comment | added | Andrew Stacey | I'm leaving this as a comment as you ask for a topological characterisation. I don't know one, but I do know an algebraic characterisation. What you describe are totally convex spaces as described at ncatlab.org/nlab/show/totally+convex+space in particular, any such set is the image of a unit ball under a continuous map from a Banach space (indeed, an $\ell^1$-space). | |
| Feb 21, 2011 at 10:19 | history | asked | Pietro Majer | CC BY-SA 2.5 |