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Concentration of the distance in finite dimensional normed spaces
Author(s) -
Arias-de-Reyna Juan,
Ball Keith,
Villa Rafael
Publication year - 1998
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300014182
Subject(s) - mathematics , unit sphere , normed vector space , unit (ring theory) , ball (mathematics) , combinatorics , space (punctuation) , decomposition , exponential growth , discrete mathematics , pure mathematics , mathematical analysis , ecology , linguistics , philosophy , mathematics education , biology
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).