LARGE SIGNED SUBSET SUMS

We study the following question: for given d ⩾ 2 , n and k ⩽ what is largest value c ( ) such that from any set of unit vectors in R we may select with corresponding signs ±1 so their signed sum has norm at least ? The problem dual to classical vector minimization balancing questions, which have been studied over a century. give asymptotically sharp estimates general case. In several special cases, provide stronger estimates: quantity corresponds ℓ p -polarization problem, while determining equivalent estimating coherence system, case p-frame energies. Two new proofs are presented Welch bound when = + 1 . For large values n, volumetric applied obtaining fine on Studying planar case, bounds given. Finally, determine exact under some extra assumptions.