Small subset sums

Gergely Ambrus; Imre Barany; Victor Grinberg

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arXiv · 1502.04027

Small subset sums

Gergely Ambrus·Imre Barany·Victor Grinberg

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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.

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BibTeX RIS

Gergely Ambrus, Imre Barany, Victor Grinberg. 2015-02-13. Small subset sums. https://arxiv.org/abs/1502.04027

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