An extension of Elton’s $\ell _1^n$ theorem to complex Banach spaces
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- by S. J. Dilworth and Joseph P. Patterson PDF
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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 \[ \mathbb {E} \left \| \sum _{i=1}^n Z_i e_i \right \| \geq \delta n \] (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.References
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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
- MR Author ID: 58105
- 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
- Received by editor(s): October 17, 2001
- Received by editor(s) in revised form: December 11, 2001
- Published electronically: 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 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1489-1500
- MSC (2000): Primary 46B07; Secondary 46B04, 46B09
- DOI: https://doi.org/10.1090/S0002-9939-02-06651-0
- MathSciNet review: 1949879