Timeline for Maximum on unit ball (James' theorem).
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Sep 2, 2010 at 15:00 | comment | added | Pietro Majer | (with $\epsilon$ large enough) | |
| Sep 2, 2010 at 14:33 | comment | added | Pietro Majer | Indeed for every Banach space $X$ and every bounded linear functional $f$, there is an equivalent norm of $X$ producing a unit ball such that $f$ has multiple maximum points there (just take $\max\{\|x\|,\epsilon |\langle f,x\rangle |\}$) | |
| Sep 2, 2010 at 14:06 | comment | added | Bill Johnson | Uniqueness is equivalent to strict convexity of the ball, Piero: If $f$ attains its max at two points $x$ and $y$, then it attains it at everything on the line segment joining $x$ and $y$. On the other hand, if the sphere contains a line segment, you can define a norm one linear functional $f$ on the two dimensional subspace spanned by the endpoints of the segment s.t. the segment is contained in $[f=1]$ and use Hahn-Banach to extend the functional to the entire space. | |
| Sep 2, 2010 at 12:16 | comment | added | Piero D'Ancona | Uniqueness seems to fail even in finite dimensional cases (think of $R^2$ with sup norm) and might be related to uniform convexity -- just a guess | |
| Sep 2, 2010 at 11:22 | history | edited | Jonas T | CC BY-SA 2.5 |
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| Sep 2, 2010 at 11:16 | history | edited | Jonas T | CC BY-SA 2.5 |
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| Sep 2, 2010 at 11:11 | comment | added | Jonas T | This seems to be James' article about this: springerlink.com/content/07411341j5792482 | |
| Sep 2, 2010 at 11:05 | comment | added | Andrew Stacey | +1 not particularly for the question but because that theorem is something I'd been wondering about recently but didn't know that someone had proven it! Don't happen to have a reference for it, do you? | |
| Sep 2, 2010 at 10:48 | vote | accept | CommunityBot | moved from User.Id=5295 by developer User.Id=481663 | |
| Sep 2, 2010 at 10:45 | answer | added | Pietro Majer | timeline score: 9 | |
| Sep 2, 2010 at 10:35 | history | asked | Jonas T | CC BY-SA 2.5 |