Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T07:16:11.405Z Has data issue: false hasContentIssue false

EQUILATERAL SETS IN UNIFORMLY SMOOTH BANACH SPACES

Published online by Cambridge University Press:  02 January 2014

D. Freeman
Affiliation:
Department of Mathematics and Computer Science, Saint Louis University, St Louis, MO 63103,U.S.A. email [email protected]
E. Odell
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-0257,U.S.A. email [email protected]
B. Sari
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-5017,U.S.A. email [email protected]
Th. Schlumprecht
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368,U.S.A. email [email protected]
Get access

Abstract

Let $X$ be an infinite-dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exist a constant $\lambda \gt 0$ and an infinite sequence $\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$ such that $\Vert {x}_{i} - {x}_{j} \Vert = \lambda $ for all $i\not = j$.

Type
Research Article
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

Alon, N. and Pudlák, P., Equilateral sets in ${ \ell }_{p}^{n} $. Geom. Funct. Anal. 13 (3) (2003), 467482.CrossRefGoogle Scholar
Guerre-Delabriere, S., Classical sequences in Banach spaces, Dekker (New York, 1992).Google Scholar
Mercourakis, S. K. and Vassiliadis, G., Equilateral sets in infinite dimensional Banach spaces. Proc. Amer. Math. Soc. 142 (2014), 205212.Google Scholar
Odell, E., Stability in Banach spaces. Extracta Math. 17 (3) (2002), 385425.Google Scholar
Petty, C. M., Equilateral sets in Minkowski spaces. Proc. Amer. Math. Soc. 29 (1971), 369374.Google Scholar
Smyth, C., Equilateral sets in ${ \ell }_{p}^{d} $. In Thirty Essays on Geometric Graph Theory (ed. Pach, J.), Springer (New York, 2013), 483488.Google Scholar
Swanepoel, K. and Villa, R., A lower bound for the equilateral number of normed spaces. Proc. Amer. Math. Soc. 136 (2008), 127131.Google Scholar
Terenzi, P., Successioni regolari negli spazi di Banach. Milan J. Math. 57 (1) (1987), 275285.Google Scholar
Terenzi, P., Equilater sets in Banach spaces. Boll. Unione Mat. Ital. A (7) 3 (1) (1989), 119124.Google Scholar