Large signed subset sums

TL;DR

This paper investigates the maximum norm of signed sums of vectors in Euclidean space, providing asymptotically sharp estimates and new proofs for classical bounds, with special focus on the planar case and specific configurations.

Contribution

It offers asymptotically sharp estimates for the largest signed sum norms, introduces new proofs for the Welch bound, and provides exact values in certain special cases.

Findings

Asymptotically sharp estimates for $c(d,n,k)$ in general.

New proofs for the classical Welch bound when $n=d+1$.

Sharp bounds on $c(2,n,k)$ in the planar case.

Abstract

We study the following question: for given $d\geq 2$, $n\geq d$ and $k \leq n$, 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.