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

The well-studied vector balancing constant β(U, V) of a pair of convex bodies $(U,V)$, is lower bounded by a lattice counterpart, α(U,V). In [BS97], Banaszczyk and Szarek proved that α(B₂ⁿ, V)≤ c when V has Gaussian measure at least 1/2, and conjectured that, for centrally symmetric V, β(B₂ⁿ, 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 α(B₂ⁿ, V) even without the central symmetry assumption.