  A tantalizing conjecture in discrete mathematics is the one of Koml\'os,
suggesting that for any vectors $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_2^m$
there exist signs $x_1, \dots, x_n \in \{ -1,1\}$ so that $\|\sum_{i=1}^n
x_i\mathbf{a}_i\|_\infty \le O(1)$. It is a natural extension to ask what
$\ell_q$-norm bound to expect for $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_p^m$.
We prove that, for $2 \le p \le q \le \infty$, such vectors admit fractional
colorings $x_1, \dots, x_n \in [-1,1]$ with a linear number of $\pm 1$
coordinates so that $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_q \leq
O(\sqrt{\min(p,\log(2m/n))}) \cdot n^{1/2-1/p+ 1/q}$, and that one can obtain a
full coloring at the expense of another factor of $\frac{1}{1/2 - 1/p + 1/q}$.
In particular, for $p \in (2,3]$ we can indeed find signs $\mathbf{x} \in \{
-1,1\}^n$ with $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_\infty \le O(n^{1/2-1/p} \cdot
\frac{1}{p-2})$. Our result generalizes Spencer's theorem, for which $p = q =
\infty$, and is tight for $m = n$.
  Additionally, we prove that for any fixed constant $\delta>0$, in a centrally
symmetric body $K \subseteq \mathbb{R}^n$ with measure at least $e^{-\delta n}$
one can find such a fractional coloring in polynomial time. Previously this was
known only for a small enough constant -- indeed in this regime classical
nonconstructive arguments do not apply and partial colorings of the form
$\mathbf{x} \in \{ -1,0,1\}^n$ do not necessarily exist.
