Timeline for Perturbing the approximation property from the Lipschitz-free space to stay in the Wasserstein space
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Dec 2, 2021 at 12:57 | comment | added | ABIM | @NikWeaver No worries; but it's also a pleasure to discuss with you (especially after reading you book/some of your papers) | |
| Dec 1, 2021 at 21:15 | comment | added | Nik Weaver | Sure. Sorry I can't be more helpful about your main question ... | |
| Dec 1, 2021 at 19:22 | comment | added | ABIM | @NikWeaver Ah, you're right; this is very simple actually. Thanks. | |
| S Dec 1, 2021 at 19:20 | history | suggested | Dirk Werner | CC BY-SA 4.0 |
typo in title
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| Dec 1, 2021 at 18:18 | review | Suggested edits | |||
| S Dec 1, 2021 at 19:20 | |||||
| Dec 1, 2021 at 13:21 | comment | added | Nik Weaver | No, the problem is that if $X$ is unbounded then you can have probability measures whose integral against some Lipschitz function is infinite. | |
| Dec 1, 2021 at 2:54 | comment | added | ABIM | As a final thought, on my end, if we equip $\mathcal{F}(X)$ with a Borel probability measure $\nu$, then since $K$ is compact then one can combine the measurable maximum theorem (18.19 in Charalambos et al.'s book) and Lusin's Theorem to conclude that a metric projection must exist on a closed subset of $\mathcal{F}(X)$ of arbitrarily-high $\nu$-probability. (Though, this is kind of cheating). | |
| Dec 1, 2021 at 2:40 | comment | added | ABIM | @NikWeaver Fair, initially I was thinking of $\epsilon$-best projections (like in Respov's book; Chapter 6) but this also seemed difficult and a likely less explored research question by others. | |
| Dec 1, 2021 at 2:39 | comment | added | ABIM | @NikWeaver Ins't the modification just $\mu \mapsto \mu - \delta_{x}$ (for the distinguishe point $x$)? | |
| Dec 1, 2021 at 2:34 | comment | added | Nik Weaver | BTW if $X$ isn't compact then you don't get $\mathcal{W}(X) \subset \mathcal{F}(X)$, but you can slightly modify the definition of $\mathcal{W}(X)$ so that this works. This is probably well-known, but anyway it's done in this paper. | |
| Dec 1, 2021 at 2:32 | comment | added | Nik Weaver | As far as I know, exact projections onto nearest points of convex subsets requires reflexivity, which you never have for $\mathcal{F}(X)$ when $X$ is infinite. What you're asking for is weaker, but still seems unlikely to me. | |
| Nov 30, 2021 at 21:36 | history | edited | ABIM | CC BY-SA 4.0 |
deleted 12 characters in body; edited title
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| Nov 30, 2021 at 21:14 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Nov 30, 2021 at 20:48 | history | asked | ABIM | CC BY-SA 4.0 |