To the Steinitz lemma in coordinate form (Q1357729)

We consider an arbitrary finite family of vectors in \(m\)-dimensional space with sum zero and the absolute value of any vector's coordinate at most 1. We prove that there exists an order of vector summation for this family such that every partial sum has the absolute value of the first coordinate at most 1 while the other coordinates are bounded by a constant at most \(O(m^{3/2})\). We also describe a polynomial-time algorithm which computes an order of vector summation for which all partial sums meet somewhat weaker bounds (at most \(O(m^2)\)) on the absolute value of coordinates from 2 to \(m\). (The absolute value of the first coordinate still remains at most 1.).