Large signed subset sums

Abstract

We study the following question: for given $d\ge 2$, $n\ge d$ and $k\le n$, what is the largest value $c(d,n,k)$ such that from any set of $n$ unit vectors in ${\mathbb{R}}^d$, we may select $k$ vectors with corresponding signs $\pm 1$ so that their signed sum has norm at least $c(d,n,k)$? The problem is dual to classical vector sum minimization and balancing questions, which have been studied for over a century. We give asymptotically sharp estimates for $c(d,n,k)$ in the general case. In several special cases, we provide stronger estimates: the quantity $c(d,n,n)$ corresponds to the ${\ell }_p$-polarization problem, while determining $c(d,n,2)$ is equivalent to estimating the coherence of a vector system, which is a special case of $p$-frame energies. Two new proofs are presented for the classical Welch bound when $n=d+1$. For large values of $n$, volumetric estimates are applied for obtaining fine estimates on $c(d,n,2)$. Studying the planar case, sharp bounds on $c(2,n,k)$ are given. Finally, we determine the exact value of $c(d,d+1,d+1)$ under some extra assumptions.

15 pages. Updated from the printed version: planar bounds are slightly corrected, and reference to sharp bound on c(2,n,n) is added