  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.
