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!
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 $$||\sum a_je_j||\le c.$$