A note on norms of signed sums of vectors

Abstract

Abstract Improving a result of Hajela, we show for every function f with lim n→∞ f(n) = ∞ and f(n) = o(n) that there exists n 0 = n 0(f) such that for every n ⩾ n 0 and any S ⊆ {–1, 1} n with cardinality |S| ⩽ 2 n/f(n) one can find orthonormal vectors x 1, …, xn ∈ ℝ n satisfying ∥ ε 1 x 1 + ⋯ + ε n x n ∥ ∞ ⩾ c log ⁡ f ( n ) $\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+\varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all (ε 1, …, εn ) ∈ S. We obtain analogous results in the case where x 1, …, xn are independent random points uniformly distributed in the Euclidean unit ball B 2 n $\begin{array}{} \displaystyle B_2^n \end{array}$ or in any symmetric convex body, and the ℓ ∞ n $\begin{array}{} \displaystyle \ell_{\infty }^n \end{array}$ -norm is replaced by an arbitrary norm on ℝ n .