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

TL;DR

This paper proves that the Gaussian measure of a convex body bounds its maximal covering radius, confirming a conjecture relating vector balancing constants to Gaussian measure, and extends results to non-symmetric bodies.

Contribution

It confirms a conjecture that the vector balancing constant is bounded by the Gaussian measure of the convex body, regardless of symmetry or dimension.

Findings

Confirmed the conjecture for centrally symmetric bodies.

Extended the result to non-symmetric convex bodies.

Established bounds relating Gaussian measure and covering radius.

Abstract

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 $\alpha(B_2^n, V)\leq c$ 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.