Balanced partitions of vector sequences

Let d,r∈N and ∥ · ∥ be any norm on Rd. Let B denote the unit ball with respect to this norm. We show that any sequence v1, v2, … of vectors in B can be partitioned into r subsequences V1, …, Vr in a balanced manner with respect to the partial sums: For all , we have . A similar bound holds for partitioning sequences of vector sets. Both results extend an earlier one of Bárány and Grinberg [I. Bárány, V.S. Grinberg, On some combinatorial questions in finite-dimensional spaces, Linear Algebra Appl. 41 (1981) 1–9] to partitions in arbitrarily many classes.