Colorful Vector Balancing

We extend classical estimates for the vector balancing constant of $\mathbb{R}^d$ equipped with the Euclidean and the maximum norms proved in the 1980's by showing that for $p =2$ and $p=\infty$, given vector families $V_1, \ldots, V_n \subset B_p^d$ with $0 \in \sum_{i=1}^n \mathrm{conv}\, V_i$, one may select vectors $v_i \in V_i$ with $\| v_1 + \ldots + v_n \|_2 \leq \sqrt{d}$ for $p=2$, and $\| v_1 + \ldots + v_n \|_\infty \leq O(\sqrt{d})$ for $p = \infty$. These bounds are sharp and asymptotically sharp, respectively, for $n \geq d$. The proofs combine linear algebraic and probabilistic methods with a Gaussian random walk argument.