Vector balancing in Lebesgue spaces

The Komlós conjecture suggests that for any vectors a1,…,an∈B2m$$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_2^m $$ there exist x1,…,xn∈{−1,1}$$ {x}_1,\dots, {x}_n\in \left\{-1,1\right\} $$ so that ‖∑i=1nxiai‖∞≤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,…,an∈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 e−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.