Small subset sums

TL;DR

This paper proves bounds on the norm of subset sums in finite sets within a normed space, providing sharp bounds and improvements for Euclidean and max norms, with applications to vector sums in the plane.

Contribution

It establishes new bounds on subset sums in normed spaces, including sharp bounds and improvements for specific norms, and applies these results to planar vector sums.

Findings

Existence of subset sums with bounded norm for any subset size

Sharpness of the general bound in normed spaces

Improved bounds for Euclidean and max norms

Abstract

Let ||.|| be a norm in R^d whose unit ball is B. Assume that V\subset B is a finite set of cardinality n, with \sum_{v \in V} v=0. We show that for every integer k with 0 \le k \le n, there exists a subset U of V consisting of k elements such that \| \sum_{v \in U} v \| \le \lceil d/2 \rceil. We also prove that this bound is sharp in general. We improve the estimate to O(\sqrt d) for the Euclidean and the max norms. An application on vector sums in the plane is also given.