Timeline for Geometric implications of $\beta(B_X) = 2$
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 8, 2016 at 17:05 | comment | added | Bunyamin Sari | As these example show, the question should be really what if $\beta=2$ under all renormings. I still think this may imply non-reflexivity and if you are interested you may have to dig out the proof from [OS] paper i mentioned in the first answer. For instance, Corollary 3.5 of the paper seems to imply the unconditionality concern I had before isn't an issue. | |
| Dec 8, 2016 at 16:32 | comment | added | anonymous | @BillJohnson Thank you, I understand now (the typo had confused me greatly). Indeed it's clear that this gives a norm $\phi$ that is equivalent to the canonical $\ell^2$ norm yet $\phi(e_n - e_m) = 4$ whenever $n \ne m$ and $\phi(e_n) = 2$ for every $n$. That's a very pleasantly simple example. | |
| Dec 8, 2016 at 16:11 | comment | added | Bill Johnson | @anonymous. See my second comment after my answer and correct the typo (non reflexivity instead of reflexivity). | |
| S Dec 8, 2016 at 14:32 | history | edited | David Handelman | CC BY-SA 3.0 |
Correct the title of the reference (about -> on), add more info, in particular an MR link, and (hope this is not too invasive) make it stand out a bit; added verb to sentence [dh].
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| S Dec 8, 2016 at 14:32 | history | suggested | anonymous | CC BY-SA 3.0 |
Correct the title of the reference (about -> on), add more info, in particular an MR link, and (hope this is not too invasive) make it stand out a bit.
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| Dec 8, 2016 at 14:27 | review | Suggested edits | |||
| S Dec 8, 2016 at 14:32 | |||||
| Dec 8, 2016 at 13:59 | comment | added | anonymous | Interesting. So if I understand correctly, not even the stronger version, where the supremum in the definition of $\beta(B_X) = 2$ is attained (which is what one actually has in $\ell^1$, $\ell^\infty$, etc.) suffices to guarantee non-reflexivity. | |
| Dec 8, 2016 at 6:47 | vote | accept | anonymous | ||
| Dec 7, 2016 at 23:45 | history | edited | Bunyamin Sari | CC BY-SA 3.0 |
added 264 characters in body
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| Dec 7, 2016 at 22:46 | comment | added | Bunyamin Sari | What they actually prove is: if under all renorming $X$ admits a spreading model $(e_i)$ with $\|e_1\pm e_2\|=2$, then $\ell_1$ embeds in $X$. So I should have said if $\beta_{|\cdot|}(S_X)=2$ for all renorming $|\cdot|$ of $X$. Even so this is still not quite enough, will need unconditionality and $\beta=2$ need to be attained. Not sure at the moment if one can overcome these. | |
| Dec 7, 2016 at 21:34 | comment | added | Bill Johnson | How does it follow from [OS]? | |
| Dec 7, 2016 at 21:00 | history | answered | Bunyamin Sari | CC BY-SA 3.0 |