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Conjecture. Let $n\in\mathbb{N}$, and let $(e_i)_{i=1}^n$ be a normalized unconditional basis for an $n$-dimensional Banach space $E_n$ with Schauder basis constant $K\in[1,\infty)$. If $\|e_1+e_2+\cdots+e_n\|\leq C$ then $(e_i)_{i=1}^n$ is $2CK$-equivalent to the canonical basis of $\ell_\infty^n$.

I seem to remember that there is something like this in infinite dimensions, but I am concerned about the finite-dimensional conjecture above. Is it true? If so, is there a reference?

The constant $2CK$ is my best guess. Really, I just want any equivalence constant $\kappa=\kappa(C,K)$ which is a function of $C$ and $K$ but independent of $n$.

Thanks!

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    $\begingroup$ In $\ell_1^n$, take $x_1=e_1, x_2=e_2-e_1,x_3=e_3-e_2,\ldots, x_n= e_n-{e_{n-1}}$. This is a monotone basis, with $\Vert x_1+\cdots +x_n\Vert=1$. One also has $\Vert x_1-x_2+\cdots+(-1)^{n} x_n\Vert_\infty=2$ and $\Vert x_1-x_2+\cdots+(-1)^{n} x_n\Vert_1=2n-1$. $\endgroup$ – David Mitra Jul 11 '16 at 18:46
  • $\begingroup$ I think it's clearly yes, at least with a constant something like $2CK$, if you assume $||\pm e_1\pm\dots\pm\pm e_n||\le C$. It seems implausible without the $\pm$ signs, not that I have a counterexample. $\endgroup$ – David C. Ullrich Jul 11 '16 at 18:46
  • $\begingroup$ Ah okay, let's assume the basis is unconditional, then. $\endgroup$ – Ben W Jul 11 '16 at 18:48
  • $\begingroup$ You want to bound the unconditional basis constant; any basis of a finite dimensional space is unconditional. $\endgroup$ – David Mitra Jul 11 '16 at 18:56
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If the basis is unconditional (with, as David Mitra points out, the unconditional basis constant also bounded by something independent of $n$) then yes, at least with some constant independent of $n$. Using $c$ to denote anything not depending on $n$:

Of course $|a_k|\le c||\sum a_je_j||$ just because it's a normalized basis.

On the other hand, if $-1\le a_j\le 1$ then $\sum a_je_j$ is a convex combination of vectors of the form $\sum\pm e_j$, so $$||\sum a_je_j||\le c.$$

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  • $\begingroup$ Great, thanks! By the way, do you know if there are any references for this? I seem to recall a paper from 1975 by Altshuler showing that certain Lorentz sequence spaces contain $\ell_\infty^n$ uniformly, and hence are not uniformly convexifiable. Probably he uses this fact, and so a reference should be there. However, I don't currently have access to Israel J Math. $\endgroup$ – Ben W Jul 11 '16 at 19:10
  • $\begingroup$ @BenWallis I know nothing about Banach spaces, sorry - had to look up "unconditional" to see what it had to do with my version of the revised hypothesis. I bet Mitra knows... $\endgroup$ – David C. Ullrich Jul 11 '16 at 19:13
  • $\begingroup$ Ah, no problem. The important thing is that the conjecture is true for unconditional bases. Thank you! $\endgroup$ – Ben W Jul 11 '16 at 19:14
  • $\begingroup$ @BenWallis, this is attributed to Bill Johnson.You will find (a form of) it in the Elton--Odell paper about separation in Banach spaces. $\endgroup$ – Tomek Kania Jul 11 '16 at 19:19

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