Let $X$ be a Banach space. We say that $X$ contains $\ell\_1^n$'s uniformly iff for all $n\in\mathbb N$ there exist subspaces $X\_n\subseteq X$ with $d(X\_n,\ell\_1^n)\leq \lambda$ for some $\lambda\geq 1$. A famous theorem of Pisier's asserts that an infinite dimensional Banach space is $K$-convex iff it does not contain $\ell\_1^n$'s uniformly. For the notion of $K$-convexity look at <https://bookstore.ams.org/cbms-60>. I want to know the following suppose an infinite dimensional Banach space $X$ is not $K$-convex. Now assume that $X$ is an inductive limit of Banach spaces $Y\_1\subseteq Y\_2\subseteq\dots\subseteq Y\_n\subseteq\dots$ where $\text{dim}Y\_n=n.$ Let $X$ not $K$-convex. Clearly without loss of generality we may assume that there exists an increasing sequence $(k\_n)$ such that $Y\_n$ contains a subspace of dimension $k\_n$ which is $\lambda$-isomorphic to $\ell\_1^{k\_n}$ for all $n\geq 1$ and $\lambda\geq 1$ fixed positive constant. My question is if there are any known estimates on the norm of projection from $Y\_n$ to the subspace which is $\lambda$-isomorphic to $\ell\_1^{k\_n}$?



$K$-convex Banach spaces
$Y\_n$ can be $\ell\_\infty^n$, in which case the best projection onto any $\ell\_1^k$ is of order at least $\sqrt{k}$.


