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ON THE EXISTENCE OF 1-SEPARATED SEQUENCES ON THE UNIT BALL OF A FINITE-DIMENSIONAL BANACH SPACE

Part of: Discrete geometry General convexity Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  05 December 2014

E. Glakousakis  and
S. Mercourakis
E. Glakousakis
Affiliation:
University of Athens, Department of Mathematics, Panepistimioupolis, 15784 Athens, Greece email e.glakousakis@gmail.com
S. Mercourakis
Affiliation:
University of Athens, Department of Mathematics, Panepistimioupolis, 15784 Athens, Greece email smercour@math.uoa.gr
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Abstract

Given a finite-dimensional Banach space $X$ and an Auerbach basis $\{(x_{k},x_{k}^{\ast }):1\leqslant k\leqslant n\}$ of $X$, it is proved that there exist $n+1$ linear combinations $z_{1},\ldots ,z_{n+1}$ of $x_{1},\ldots ,x_{n}$ with coordinates $0,\pm 1$, such that $\Vert z_{k}\Vert =1$, for $k=1$, $2,\ldots ,n+1$ and $\Vert z_{k}-z_{l}\Vert >1$, for $1\leqslant k<l\leqslant n+1$.

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
Secondary: 52C99: None of the above, but in this section

Information

Type
Research Article
Information
Mathematika , Volume 61 , Issue 3 , September 2015 , pp. 547 - 558
Copyright
Copyright © University College London 2014 

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References

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