Timeline for Convex hulls of families of probability measures
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2023 at 16:00 | history | edited | Glorfindel | CC BY-SA 4.0 |
broken link fixed, cf. https://meta.mathoverflow.net/q/5301/70594
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| Mar 3, 2016 at 10:25 | vote | accept | SBF | ||
| S Dec 12, 2015 at 13:23 | history | suggested | Ali Taghavi |
I add a tag
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| Dec 12, 2015 at 12:52 | review | Suggested edits | |||
| S Dec 12, 2015 at 13:23 | |||||
| Jan 1, 2014 at 21:06 | answer | added | 7891user | timeline score: 4 | |
| Jan 1, 2014 at 17:32 | comment | added | SBF | @D.Kelleher: sorry, I misread your previous comment - now I understand what you meant, thanks for clarifying. The idempotence of $\operatorname{sco}$ holds at least for analytic sets (see my answer below). | |
| Jan 1, 2014 at 17:30 | answer | added | SBF | timeline score: 2 | |
| Jan 1, 2014 at 16:25 | comment | added | D. Kelleher | @Ilya Ahh, I'll clarify. The Choquet theorem doesn't use idempotence of $\operatorname{sco}$. What it says is that Convex sets are exactly the Range of $\operatorname{sco}$. My suggestion was: if we knew that $\operatorname{sco}$ is idempotent, the Choquet theorem implies that convex sets are all of the fixed points of $\operatorname{sco}$. But since I don't know for a fact this is true, I didn't feel right calling it an "answer." | |
| Jan 1, 2014 at 14:35 | comment | added | SBF | @D.Kelleher: I see, could you clarify where the assumption of the idempotence of $\operatorname{sco}$ is used in Choquet's theorem? I don't see it is mentioned in the formulation of the result. | |
| Dec 28, 2013 at 15:06 | comment | added | D. Kelleher | @Ilya The Choquet theorem is on the wiki page (though it should be in "Lecture notes on" too), and essentially says that a convex set is $\operatorname{sco} P$ for some $P$. I would think that it should be idempotent pretty generally, but i don't know off hand, and generalizing the proof from convex combinations runs into measure issues. | |
| Dec 28, 2013 at 13:03 | comment | added | SBF | @D.Kelleher: can you suggest any sufficient conditions for the idempotence of $\operatorname{sco}$? Also, the Choquet's theorem you are talking about can be found in "Lecture notes on" I guess, isn't it? | |
| Dec 28, 2013 at 4:03 | answer | added | Gerald Edgar | timeline score: 3 | |
| Dec 27, 2013 at 22:09 | comment | added | D. Kelleher | The study of this sort of convex hull is the domain of Choquet theory. In fact assuming $\operatorname{sco}$ is idempotent, the Choquet theorem implies that convex sets are exactly the fixed points of $\operatorname{sco}$. The wiki page has some good references en.wikipedia.org/wiki/Choquet_theory | |
| Dec 27, 2013 at 18:51 | comment | added | SBF | @MichaelGreinecker: by a convex combination I mean the integral over a "weighting" measure $\nu$ which gives a full outer measure to $P$ (at least I'm not aware of any other definition to be used here). Can you say that $\operatorname{sco}$ is idempotent? | |
| Dec 27, 2013 at 17:40 | comment | added | Michael Greinecker | Why is this "exactly the set of all convex combinations"? And wouldn't that answer the question since taking convex hulls is an idempotent operation? | |
| Dec 27, 2013 at 15:58 | history | edited | SBF | CC BY-SA 3.0 |
added 210 characters in body
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| Dec 27, 2013 at 13:48 | history | asked | SBF | CC BY-SA 3.0 |