A matrix version of the Steinitz lemma

The Steinitz lemma, a classic from 1913, states that a(1), ... , a(n), a sequence of vectors in R-d with Sigma(n)(i=1) a(i) = 0, can be rearranged so that every partial sum of the rearranged sequence has norm at most 2d max parallel to a(i)parallel to. In the matrix version A is a k x n matrix with entries a(i)(j) is an element of R-d with Sigma(k)(j=1) Sigma(n)(i=1) a(i)(j) = 0. It is proved in [T. Oertel, J. Paat and R. Weismantel, A colorful Steinitz lemma with applications to block integer programs, Math. Program. 204 (2024), 677-702] that there is a rearrangement of row j of A (for every j) such that the sum of the entries in the first m columns of the rearranged matrix has norm at most 40d(5) max parallel to a(i)(j)parallel to (for every m). We improve this bound to (4d - 2) max parallel to a(i)(j)parallel to.