Vector Balancing in Lebesgue Spaces
Victor Reis, Thomas Rothvoss

TL;DR
This paper extends vector balancing results in Lebesgue spaces, providing new bounds for fractional and full colorings in high-dimensional spaces, generalizing Spencer's theorem and offering polynomial-time algorithms.
Contribution
It introduces bounds for vector colorings in Lebesgue spaces for a range of p and q, generalizing Spencer's theorem and providing constructive algorithms.
Findings
Fractional colorings with linear number of ±1 coordinates are achievable.
Bounds depend on p, q, and the dimension, with explicit formulas.
Polynomial-time algorithms exist for certain measure conditions in symmetric bodies.
Abstract
A tantalizing conjecture in discrete mathematics is the one of Koml\'os, suggesting that for any vectors there exist signs so that . It is a natural extension to ask what -norm bound to expect for . We prove that, for , such vectors admit fractional colorings with a linear number of coordinates so that , and that one can obtain a full coloring at the expense of another factor of . In particular, for we can indeed find signs with $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_\infty \le O(n^{1/2-1/p}…
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Advanced Mathematical Modeling in Engineering · Mathematical Approximation and Integration
