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Concentration of the distance in finite dimensional normed spaces

Published online by Cambridge University Press:  26 February 2010

Juan Arias-de-Reyna ,
Keith Ball  and
Rafael Villa
Juan Arias-de-Reyna
Affiliation:
Depto. Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia, S/N, 41012 Sevilla, Spain. e-mail: arias@cica.es
Keith Ball
Affiliation:
Department of Mathematics, University College London, Gower Street, London WCIE 6BT. e-mail: kmb@math.ucl.ac.uk
Rafael Villa
Affiliation:
Depto. Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia, S/N, 41012 Sevilla, Spain. e-mail: rvcaro@cica.es
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Abstract

We prove that in every finite dimensional normed space, for “most” pairs (x, y) of points in the unit ball, ║x − y║ is more than √2(1 − ε). As a consequence, we obtain a result proved by Bourgain, using QS-decomposition, that guarantees an exponentially large number of points in the unit ball any two of which are separated by more than √2(1 − ε).

Type
Research Article
Information
Mathematika , Volume 45 , Issue 2 , December 1998 , pp. 245 - 252
Copyright
Copyright © University College London 1998

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