An extension of Elton’s $\ell _1^n$ theorem to complex Banach spaces

Abstract:

Let $\varepsilon > 0$ be sufficiently small. Then, for $\theta =0.225\sqrt \varepsilon$, there exists $\delta := \delta (\varepsilon )<1$ such that if $(e_i)_{i=1}^n$ are vectors in the unit ball of a complex Banach space $X$ which satisfy \[ \mathbb {E} \left \| \sum _{i=1}^n Z_i e_i \right \| \geq \delta n \] (where $(Z_i)$ are independent complex Steinhaus random variables), then there exists a set $B \subseteq \{1,\dots ,n\}$, with $|B| \geq \theta n$, such that \begin{equation*} \left \|\sum _{i\in B} z_i e_i \right \| \geq (1-\varepsilon ) \sum _{i\in B} |z_i| \end{equation*} for all $z_i\in \mathbb {C}$ ($i\in B$). The $\sqrt \varepsilon$ dependence on $\varepsilon$ of the threshold proportion $\theta$ is sharp.