The Gaussian measure of a convex body controls its maximal covering radius

The well-studied vector balancing constant $\beta (U,V)$ of a pair of convex bodies $(U,V)$, is lower bounded by a lattice counterpart, $\alpha (U,V)$. In [BS97], Banaszczyk and Szarek proved that $\alpha (B_2^n,V)\le c$ when $V$ has Gaussian measure at least $\frac{1}{2}$, and conjectured that, for centrally symmetric $V$, $\beta (B_2^n,V)$ is always bounded by a function of the Gaussian measure of $V$, independent of $n$. We resolve this conjecture in the affirmative. Moreover, we show that the analogous result holds for $\alpha (B_2^n,V)$ even without the central symmetry assumption.