Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T17:51:32.131Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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 [email protected]
S. Mercourakis
Affiliation:
University of Athens, Department of Mathematics, Panepistimioupolis, 15784 Athens, Greece email [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arias-de-Reyna, L., Ball, K. and Villa, R., Concentration of the distance in finite dimensional normed spaces. Mathematika 45 1998, 245–252.CrossRefGoogle Scholar
Elton, J. and Odell, E., The unit sphere of every infinite dimensional normed linear space contains a (1+𝜀)-separated sequence. Colloq. Math. 44 1981, 105–109.CrossRefGoogle Scholar
Hajek, P., Montesinos Santalucia, V., Vanderwerff, J. and Zizler, V., Biorthogonal Systems in Banach Spaces (CMS Books in Mathematics/Ouvrages de Mathematique de la SMC 26), Springer (New York, 2008).Google Scholar
Kottman, C. A., Subsets of the unit ball that are separated by more than one. Studia Math. 53 1975, 15–27.CrossRefGoogle Scholar