Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

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
MathSciNet review: 1949879
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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 which satisfy

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

(where

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.

References:

1.: N. Alon, On the density of sets of vectors, Discrete Math. $\mathbf{46}$ (1983), 199-202. MR 85b:05004
2.: B. Bollobàs, Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability, Cambridge University Press, Cambridge, 1986. MR 88g:05001
3.: W. J. Davis, D. J. H. Garling and N. Tomczak-Jaegermann, The complex convexity of quasi-normed linear spaces, J. Funct. Anal. 55 (1984), 110-150. MR 86b:46032
4.: S. J. Dilworth and Denka Kutzarova, On the optimality of a theorem of Elton on Subsystems, Israel J. Math. 124 (2001), 215-220.
5.: John Elton, Sign-embeddings of $l\sp{n}\sb{1}$ , Trans. Amer. Math. Soc. $\mathbf{279}$ (1983), 113-124. MR 84g:46023
6.: M. G. Karpovsky and V. D. Milman Coordinate density of sets of vectors, Discrete Math. $\mathbf{24}$ (1978), 177-184. MR 80m:05004
7.: A. Pajor, Plongement de dans les espaces de Banach complexes, C. R. Acad. Sci. Paris Sér. I Math $\mathbf{296}$ (1983), 741-743. MR 84g:46024
8.: A. Pajor, Sous-espaces des espaces de Banach, Travaux en Cours, Herman, Paris, 1985. MR 88h:46028
9.: N. Sauer, On the density of families of sets, J. Combinatorial Theory Ser. A $\mathbf{13}$ (1972), pp. 145-147. MR 46:7017
10.: Saharon Shelah, Stability, the f.c.p., and the superstability; model theoretic properties of formulas in first order theory, Annals of Math. Logic, pp. 271-362, 1971. MR 47:6475
11.: Saharon Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972), 247-261. MR 46:7018
12.: S. J. Szarek and M. Talagrand, An isomorphic version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube, Geometric aspects of functional analysis (1987-1988), 105-112; Lecture Notes in Math., 1376, Springer, Berlin, 1989. MR 90h:46034
13.: M. Talagrand, Type, infratype and the Elton-Pajor theorem, Invent. Math. 107 (1992), 41-59. MR 92m:46028

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B07, 46B04, 46B09

Retrieve articles in all Journals with MSC (2000): 46B07, 46B04, 46B09

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

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|