Small subset sums
Metric Geometry
2020-12-04 v2
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.
Keywords
Cite
@article{arxiv.1502.04027,
title = {Small subset sums},
author = {Gergely Ambrus and Imre Barany and Victor Grinberg},
journal= {arXiv preprint arXiv:1502.04027},
year = {2020}
}
Comments
12 pages. Corrected, updated version: an important condition is added to the statement of Theorem 7