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An extension of Elton's $\ell_1^n$ theorem to complex Banach spaces

Author(s): S. J. Dilworth; Joseph P. Patterson
Journal: Proc. Amer. Math. Soc. 131 (2003), 1489-1500.
MSC (2000): Primary 46B07; Secondary 46B04, 46B09
Posted: September 5, 2002
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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

\begin{displaymath}\mathbb{E}\left\Vert \sum_{i=1}^n Z_i e_i \right\Vert \geq \delta n \end{displaymath}

(where $(Z_i)$ are independent complex Steinhaus random variables), then there exists a set $B \subseteq \{1,\dots ,n\}$, with $\vert B\vert \geq \theta n $, such that

\begin{displaymath}\left\Vert\sum_{i\in B} z_i e_i \right\Vert \geq (1-\varepsilon) \sum_{i\in B} \vert z_i\vert \end{displaymath}

for all $z_i\in \mathbb{C} $ ($i\in B$). The $\sqrt\varepsilon$ dependence on $\varepsilon$ of the threshold proportion $\theta$is sharp.

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Additional Information:

S. J. Dilworth
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Address at time of publication: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: dilworth@math.sc.edu

Joseph P. Patterson
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Address at time of publication: 2110 Arrowcreek Dr., Apt. 101, Charlotte, North Carolina 28273
Email: joe_p_chess@yahoo.com

DOI: 10.1090/S0002-9939-02-06651-0
PII: S 0002-9939(02)06651-0
Received by editor(s): October 17, 2001
Received by editor(s) in revised form: December 11, 2001
Posted: September 5, 2002
Additional Notes: The research of the first author was completed while on sabbatical as a Visiting Scholar at The University of Texas at Austin.
This paper is based on the second author's thesis for his MS degree at the University of South Carolina.
Communicated by: N. Tomczak-Jaegermann
Copyright of article: Copyright 2002, American Mathematical Society

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