A note on norms of signed sums of vectors
AbstractImproving a result of Hajela, we show for every function f with limn→∞f(n) = ∞ and f(n) = o(n) that there exists n0 = n0(f) such that for every n ⩾ n0 and any S ⊆ {–1, 1}n with cardinality |S| ⩽ 2n/f(n) one can find orthonormal vectors x1, …, xn ∈ ℝn satisfying $\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+\varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all (ε1, …, εn) ∈ S. We obtain analogous results in the case where x1, …, xn are independent random points uniformly distributed in the Euclidean unit ball $\begin{array}{} \displaystyle B_2^n \end{array}$ or in any symmetric convex body, and the $\begin{array}{} \displaystyle \ell_{\infty }^n \end{array}$-norm is replaced by an arbitrary norm on ℝn.