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Jul
2
awarded  Curious
Jun
5
comment Characterization of $l_p$ up to a linear isometry
Ah, I see... You mean that the unit ball (of the norm) must be a polytope, and in addition it must have $2n$ facets. Is this true? Anyway, this looks too complicated... Is there a way to describe the situation without the additional assumption that the inut ball is a polytope?
Jun
5
comment Characterization of $l_p$ up to a linear isometry
Mark, if we take this definition of facet, then for an absorbing bounded balanced convex set $B$ in ${\mathbb R}^3$ having 6 facets is not equivalent to being a unit ball for a space isometrically isomorphic to $\ell_\infty^3$. You can take $B=\{x\in {\mathbb R}^3: \ \max_i|x_i|\le 1\quad \&\quad x_1^2+x_2^2+x_3^2\le 2\}$ (the intersection of a cube with the edge 2 and the usual ball of radius $\sqrt{2}$) -- this set has 6 facets but it is not a cube.
Jun
5
comment Characterization of $l_p$ up to a linear isometry
@Mark Meckes: is there a direct way to define facet, other than "extreme point of the polar"?
Jun
3
revised A decomposition of a representation via characters of a normal compact subgroup
added 24 characters in body
Jun
3
revised A decomposition of a representation via characters of a normal compact subgroup
added 599 characters in body
Jun
2
asked A decomposition of a representation via characters of a normal compact subgroup
May
26
comment Example of locally convex space such that its weak and initial topology coincide
I gave an answer here: mathoverflow.net/questions/156540/…
May
26
comment Sufficient condition such that weak and initial topology coincide for a locally convex space
This is strange for me. What is the connection between the finest and the weak topologies on $X$?
May
26
answered Sufficient condition such that weak and initial topology coincide for a locally convex space
May
25
comment Characterization of $l_p$ up to a linear isometry
Bill, that idea with facets, did it get any formalization in the question about $\ell_\infty$?
May
25
comment Characterization of $l_p$ up to a linear isometry
Ah, OK, thank you. Do your words mean that infinite dimensional $\ell_\infty$ were not characterised?
May
25
comment Characterization of $l_p$ up to a linear isometry
Bill, before I find this reading, what is "1 symmetric basis"?
May
25
comment Characterization of $l_p$ up to a linear isometry
And what about other $\ell_p$?
May
25
comment Characterization of $l_p$ up to a linear isometry
Bill, first, excuse me, what is $\ell_\infty^*$? And that is interesting, of course, but it sounds not quite geometrically, more in terms of category theory.
May
25
comment Characterization of $l_p$ up to a linear isometry
@Mark Meckes: I think, this depends on the definition of facet. And I have no idea about $p\notin\{1,2,\infty\}$, I thought this must be digged up.
May
25
revised Characterization of $l_p$ up to a linear isometry
specification
May
25
comment Characterization of $l_p$ up to a linear isometry
For example, as far as I understand, the following statement is true: a Banach space $X$ is isometrically isomorphic to $\ell_1^n$, $n\in{\mathbb N}$, if and only if $\dim X=n$, and the unit ball of $X$ has exactly $2n$ extreme points. Is something similar true, say, for $\ell_\infty^n$?
May
25
comment Characterization of $l_p$ up to a linear isometry
Bill, of course I mean another thing. The question is which geometric properties of (the unit ball in) a Banach space $X$ allow us to conclude that $X$ is isometrically isomorphic to $\ell_p^n$. And I don't ask about Hilbert spaces.
May
25
comment Characterization of $l_p$ up to a linear isometry
Bill, this classification where is it described? And what is not clear in my question?