Random Vectors in the Isotropic Position

Abstract Letybe a random vector in R n, satisfying E y⊗y=id. LetMbe a natural number and lety1, …, yMbe independent copies ofy. We study the question of approximation of the identity operator by finite sums of the tensorsyi⊗yi. We prove that for some absolute constantC E 1 M ∑ i=1 M y i ⊗y i −id ⩽C· log n M ·( E ‖y‖ log M ) 1/log M , provided that the last expression is smaller than 1. We apply this estimate to improve a result of Bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position.

• Publication year

  1996

• Venue

  Journal of Functional Analysis

• Publication date

  1996-08-19

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1006/JFAN.1998.3384arXiv math/9608208
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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