On series of signed vectors and their rearrangements

Let x 1 ,…, x m ∈ \input amssym $ \Bbb R$ n be a sequence of vectors with ∥ x i ∥ 2 ≤ 1 for all i . It is proved that there are signs ε 1 ,…,ε m = ±1 such that\documentclass{article} \usepackage{mathrsfs,amsmath, amssymb} \pagestyle{empty} \begin{document}\begin{align*} \|{\varepsilon}_1x_1+\cdots+{\varepsilon}_kx_k\|_2\le C_1\sqrt{n}+C_2\sqrt{\mathstrut\log m}, \qquad k=1,\ldots,m, \end{align*} \end{document} where C 1 , C 2 are some numerical constants. It is also proved that there are signs ε   1 ′ ,…,ε  m ′= ±1 and a permutation π of {1,…, m } such that\documentclass{article} \usepackage{mathrsfs,amsmath, amssymb} \pagestyle{empty} \begin{document}\begin{align*} \big\|{\varepsilon}_1^\prime x_{\pi(1)}+\cdots+{\varepsilon}_k^{\prime}x_{\pi(k)}\big\|_2\le C^{\prime}\sqrt n,\qquad k=1,\ldots,m, \end{align*} \end{document} where C ′ is some other numerical constant. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011