Vector balancing in Lebesgue spaces

Abstract

The Komlós conjecture suggests that for any vectors bold-italica1,…,bold-italican∈B2m$$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_2^m $$ there exist x1,…,xn∈false{prefix−1,1false}$$ {x}_1,\dots, {x}_n\in \left\{-1,1\right\} $$ so that ‖∑i=1nxibold-italicai‖∞≤Ofalse(1false)$$ {\left\Vert {\sum}_{i=1}^n{x}_i{\boldsymbol{a}}_i\right\Vert}_{\infty}\le O(1) $$. It is a natural extension to ask what ℓq$$ {\ell}_q $$‐norm bound to expect for bold-italica1,…,bold-italican∈Bpm$$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_p^m $$. 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 δ>0$$ \delta >0 $$, a symmetric convex body K⊆ℝn$$ K\subseteq {\mathbb{R}}^n $$ with Gaussian measure at least eprefix−δn$$ {e}^{-\delta n} $$ admits a partial coloring. Previously this was known only for a small enough δ$$ \delta $$. Additionally, we show that a hereditary volume bound suffices to provide such Gaussian measure lower bounds.