Balanced Partitions of Vector Sequences

Let $d,r\in \mathbb{N}$, $\mathrm{\parallel }\cdot \mathrm{\parallel }$ any norm on ${\mathbb{R}}^d$ and $B$ denote the unit ball with respect to this norm. We show that any sequence $v_1,v_2,\mathrm{.}\mathrm{.}\mathrm{.}$ of vectors in $B$ can be partitioned into $r$ subsequences $V_1,\mathrm{.}\mathrm{.}\mathrm{.},V_r$ in a balanced manner with respect to the partial sums: For all $n\in \mathbb{N}$, $\ell \le r$,wehave$|\sum_{i \le k, v_i \in V_\ell} v_i - \tfrac 1r \sum_{i \le k} v_i| \le 2.0005 d$. A similar bound holds for partitioning sequences of vector sets. Both results extend an earlier one of B'ar'any and Grinberg (1981) to partitions in arbitrarily many classes.