| Résume | The vector balancing constant of two convex bodies $U, V \subset \R^n$, $\beta(U, V)$, is defined to be the smallest $b > 0$ such that for any $n$ vectors in $U$, some signed combination of these vectors lies in a $b$-scaled copy of $V$. This constant is the subject of the well-known Komlós conjecture, which asks whether $\beta(B_2^n, B_\infty^n)$ is bounded by a universal constant, where $B_2^n, B_\infty^n$ are the unit ball and cube, respectively, in dimension $n$. We will introduce a related parameter $\alpha(U, V) \leq \beta(U,V)$, defined via lattices, and review some known inequalities for these two parameters. In 1997, W. Banaszczyk and S. Szarek showed that (for some universal $\theta>0$) if a convex body has gaussian measure $\gamma_n(V)\geq 1/2$, then $\alpha(B_2^n, V) \leq \theta$; this yields $\alpha(B_2^n, B_\infty^n)=O(\sqrt{\log n})$ for the cube. They conjecture that a similar inequality holds for convex bodies of gaussian measure $p<1/2$, i.e., that there exists a non-increasing function $f$ (independent of $n$) such that $\beta(B_2^n, V)\leq f(\gamma_n(V))$. We answer this question in the affirmative for the parameter $\alpha$. |
