Vector balancing in Lebesgue spaces

Abstract

AbstractThe Komlós conjecture suggests that for any vectors there exist so that . It is a natural extension to ask what ‐norm bound to expect for . We prove a tight partial coloring result for such vectors, implying a nearly tight full coloring bound. As a corollary, this implies a special case of Beck–Fiala's conjecture. We achieve this by showing that, for any , a symmetric convex body with Gaussian measure at least admits a partial coloring. Previously this was known only for a small enough . Additionally, we show that a hereditary volume bound suffices to provide such Gaussian measure lower bounds.