Timeline for $\overline{conv}(C)$, where $C = \{ e _{1}, \cdots e _{n} \}$, $e _{i} \in \ell ^{p, \infty}$ is diametral
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2019 at 21:51 | vote | accept | G. P | ||
| Feb 14, 2019 at 21:00 | answer | added | Dirk Werner | timeline score: 0 | |
| Feb 14, 2019 at 19:26 | comment | added | G. P | no, exists subset of some reflexive spaces that can be compact, convex and diametrals with diameter positive. An example of that are uniformy nons-square spaces. I want prove that $\ell ^{p,\infty}$ aren't normal structure and to prove that, I want prove that $\overline{conv}(C)$ is a subset of $\ell ^{p,\infty}$ is closed, bounded, convex and diametral with positive diameter, and I want know if $\Vert x - y \Vert _{p, \infty} \leqslant 1$, because if $\Vert x - y \Vert _{p, \infty} \leqslant 1$, then $\operatorname{diam}(\overline{conv}(C)) = 1$. | |
| Feb 14, 2019 at 19:10 | comment | added | Dirk Werner | Couldn't you argue that $C$ is not diametral because it's compact? | |
| Feb 14, 2019 at 19:09 | history | edited | G. P | CC BY-SA 4.0 |
added 41 characters in body
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| Feb 14, 2019 at 19:09 | comment | added | G. P | yes, each $e _{i}$ are unit vectors and $1 < p < \infty$. | |
| Feb 14, 2019 at 9:28 | comment | added | Dirk Werner | But what are the $e_i$??? What if $n=1$ and $e_1$ is some vector of norm 1000?? [I half guess that you want to refer to the unit vectors; if so, please say so.] And which $p$ do you consider? | |
| Feb 13, 2019 at 22:37 | comment | added | G. P | Yes, I want to know if $\Vert x - y \Vert _{p, \infty} \leqslant 1$ for all $x, y$ in the closed convexd hull of $C$. How $\overline{conv}(C)$ is the smallest closed and convex set that contains C, particularly $\overline{conv}(C) \subseteq B _{\ell ^{p,\infty}}$. | |
| Feb 13, 2019 at 21:07 | comment | added | Dirk Werner | Are you assuming anything about the $e_i$, like $\|e_i\|_{p,\infty}\le1$? And is your actual question whether $\|x-y\|_{p,\infty}\le 1$ for $x,y$ in the closed convex hull of $C$? | |
| Feb 13, 2019 at 5:09 | history | asked | G. P | CC BY-SA 4.0 |