Small subset sums

Let ‖.‖‖.‖ be a norm in RdRd whose unit ball is B  . Assume that V⊂BV⊂B is a finite set of cardinality n  , with ∑v∈Vv=0∑v∈Vv=0. We show that for every integer k   with 0≤k≤n0≤k≤n, there exists a subset U of V consisting of k   elements such that ‖∑v∈Uv‖≤⌈d/2⌉‖∑v∈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.