Small subset sums

Let ∥-.∥- be a norm in ℝd whose unit ball is B. Assume that V ⊂ B is a finite set of cardinality n, with Σv ∈ Vv=0. We show that for every integer k with 0≤k≤n, there exists a subset U of V consisting of k elements such that ∥Σv ∈ Uv∥-≤ ⌈d/2⌉. We also prove that this bound is sharp in general. We improve the estimate to O(√d) for the Euclidean and the max norms. An application on vector sums in the plane is also given. © 2016 Elsevier Inc. All rights reserved