A matrix version of the Steinitz lemma

The Steinitz lemma, a classic from 1913, states that a 1 , … , a n {a_{1},\ldots,a_{n}} , a sequence of vectors in ℝ d {\mathbb{R}^{d}} with ∑ i = 1 n a i = 0 {\sum_{i=1}^{n}a_{i}=0} , can be rearranged so that every partial sum of the rearranged sequence has norm at most 2 ⁢ d ⁢ max ⁡ ∥ a i ∥ {2d\max\|a_{i}\|} . In the matrix version A is a k × n {k\times n} matrix with entries a i j ∈ ℝ d {a_{i}^{j}\in\mathbb{R}^{d}} with ∑ j = 1 k ∑ i = 1 n a i j = 0 {\sum_{j=1}^{k}\sum_{i=1}^{n}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 40 ⁢ d 5 ⁢ max ⁡ ∥ a i