Vector balancing in Lebesgue spaces

The Komlos conjecture suggests that for any vectors a1, horizontal ellipsis ,an is an element of B2m$$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_2<^>m $$ there exist x1, horizontal ellipsis ,xn is an element of{-1,1}$$ {x}_1,\dots, {x}_n\in \left\{-1,1\right\} $$ so that || n-ary sumation i=1nxiai||infinity <= O(1)$$ {\left\Vert {\sum}_{i=1}<^>n{x}_i{\boldsymbol{a}}_i\right\Vert}_{\infty}\le O(1) $$. It is a natural extension to ask what lq$$ {\ell}_q $$-norm bound to expect for a1, horizontal ellipsis ,an is an element of 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 delta>0$$ \delta >0 $$, a symmetric convex body K subset of Double-struck capital Rn$$ K\subseteq {\mathbb{R}}<^>n $$ with Gaussian measure at least e-delta n$$ {e}<^>{-\delta n} $$ admits a partial coloring. Previously this was known only for a small enough delta$$ \delta $$. Additionally, we show that a hereditary volume bound suffices to provide such Gaussian measure lower bounds.