Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-17T13:58:36.619Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Keith Ball
Affiliation:
Department of Mathematics, University College London, Gower Street, London WCIE 6BT. e-mail: [email protected]
Rafael Villa
Affiliation:
Depto. Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia, S/N, 41012 Sevilla, Spain. e-mail: [email protected]
Get access

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

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

A.Alesker, S.. On the Gromov–Milman theorem regarding the concentration phenomenon on uniformly convex sphere. To appear.Google Scholar
Ba.Barthe, F.. Geometric and Functional Analysis. To appear.Google Scholar
BB.Bastero, J. and Bernues, J.. Applications of deviation inequalities on finite metric sets. Math. Nachr. 153 (1991), 3341.CrossRefGoogle Scholar
BBK.Bastero, J., Bernues, J. and Kalton, N.. Embedding -cubes in finite dimensional 1-subsymmetric spaces. Rev. Mat. Univ. Complut. Madrid, 2 (1989), 4752.Google Scholar
BPS1.Bastero, J., Pefia, A. and Schechtman, G.. Embeddings -cubes in the orbit of an element in commutative and noncommutative -spaces. Seminario Dpto. Andlisis Matematico, Univ. Complut. Madrid.Google Scholar
BPS2.Bastero, J., Pefia, A. and Schechtman, G.. Embeddings ln -cubes in low dimensional Schatten classes. Geometric Aspects of Functional Analysis (1995), 5–11.CrossRefGoogle Scholar
Be.Beckner, W.. Inequalities in Fourier analysis. Ann. of Math., 102 (1975), 159–182.CrossRefGoogle Scholar
BMW.Bourgain, J., Milman, V. and Wolfson, H.. On type of metric spaces. Trans. Amer. Math. Soc, 294 (1986), 295–317.CrossRefGoogle Scholar
BL.Brascamp, H. J. and Leib, E. H.. Best constants in Young's inequality, its converse, and its generalization to more than three functions. Advances in Math., 20 (1976), 151173.CrossRefGoogle Scholar
BRR.Burlak, J. A. C., Rankin, R. A. and Robertson, A. P.. The packing of spheres in the space 1p. Proc. Glasgow Math. Assoc, 4 (1958), 22–25.CrossRefGoogle Scholar
D.Dominguez-Benavides, T.. Some properties of the set and ball measures of noncompactness and applications. J. London Math. Soc. (2), 34 (1986), 120–128.CrossRefGoogle Scholar
EO.Elton, J. and Odell, E.. The unit ball of every infinite-dimensional normed linear space contains a(1 + ϕ)-separated sequence. Colloq. Math., 44 (1981), 105109.CrossRefGoogle Scholar
FL.Füredi, Z. and Loeb, P. A.. On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc, 121 (1994), 10631073.CrossRefGoogle Scholar
GM.Gromov, M. and Milman, V. D.. Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces. Compositio Math., 62 (1987), 263282.Google Scholar
K.Kottman, C. A.. Subsets of the unit ball that are separated by more than one. Studia Math., 53 (1975), 1527.CrossRefGoogle Scholar
M.Maurey, B.. Some deviation inequalities. Geom. Fund. Anal., 1 (1991), 188197.CrossRefGoogle Scholar
Pe.Petty, C. M.. Equilateral sets in Minkowski spaces. Proc. Amer. Math. Soc, 29 (1971). 369–374.CrossRefGoogle Scholar
S.Schmuckenschläger, M.. A concentration of measure phenomenon on uniformly convex bodies. Geometric Aspects of Functional Analysis (Israel, 1992 1994), (1995), 275–287.Google Scholar
T.Talagrand, M.. A new isoperimetric inequality and the concentration of measure phenomenon. Geometric Aspects of Functional Analysis (Israel Seminar, 1989–1990). Lecture Notes in Math., 1469 (1991), 94–124.CrossRefGoogle Scholar