Vector Balancing in Lebesgue Spaces

@article{Reis2020VectorBI,
  title={Vector Balancing in Lebesgue Spaces},
  author={Victor Reis and Thomas Rothvoss},
  journal={ArXiv},
  year={2020},
  volume={abs/2007.05634}
}
A tantalizing conjecture in discrete mathematics is the one of Komlos, 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… 
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