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Keywords:

  • balancing vectors;
  • Steinitz lemma;
  • rearrangements of series

Abstract

Let x1,…,xm\input amssym $ \Bbb R$n be a sequence of vectors with ∥xi2 ≤ 1 for all i. It is proved that there are signs ε1,…,εm = ±1 such that

  • equation image

where C1, C2 are some numerical constants. It is also proved that there are signs εmath image,…,εmath image = ±1 and a permutation π of {1,…,m} such that

  • equation image

where C is some other numerical constant. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011