Sign-embeddings of $l^{n}_{1}$

Abstract:

If $({e_i})_{i = 1}^n$ are vectors in a real Banach space with $\parallel {e_i}\parallel \leqslant 1$ and $\text {Average}_{{\varepsilon _1} = \pm 1}\parallel \sum \nolimits _{i = 1}^n {{\varepsilon _i}{e_i}\parallel \geqslant \delta n}$, where $\delta > 0$, then there is a subset $A \subseteq \{ 1,\ldots ,n\}$ of cardinality $m \geqslant cn$ such that ${({e_i})_{i \in A}}$ is $K$-equivalent to the standard $l_1^m$ basis, where $c > 0$ and $K < \infty$ depend only on $\delta$. As a corollary, if $1 < p < \infty$ and $l_1^n$ is $K$-isomorphic to a subspace of ${L_p}(X)$, then $l_1^m(m \geqslant cn)$ is $K’$-isomorphic to a subspace of $X$, where $c > 0$ and $K’ < \infty$ depend only on $K$ and $p$.