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# Euclidean Sections of Direct Sums of Normed Spaces

Published:2003-06-01
Printed: Jun 2003
• A. E. Litvak
• V. D. Milman
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## Abstract

We study the dimension of random'' Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from \cite{LMS}, to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much weaker'' randomness of diagonal'' subspaces (Corollary~\ref{sle} and explanation after). We also add some relative information on phase transition''.
 Keywords: Dvoretzky theorem, random'' Euclidean section, phase transition in asymptotic convexity
 MSC Classifications: 46B07 - Local theory of Banach spaces 46B09 - Probabilistic methods in Banach space theory [See also 60Bxx] 46B20 - Geometry and structure of normed linear spaces 52A21 - Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]