The lower bound for Koldobsky's slicing inequality via random rounding

Abstract

We study the lower bound for Koldobsky's slicing inequality. We show that there exists a measure $\mu$ and a symmetric convex body $K\subseteq {\mathbb{R}}^n$, such that for all $\xi \in S^{n-1}$ and all $t\in \mathbb{R},$ $${\mu }^{+}(K\cap ({\xi }^{\perp }+t\xi ))\le \frac{c}{\sqrt{n}}\mu (K)|K{|}^{-\frac{1}{n}}.$$ Our bound is optimal, up to the value of the universal constant. It improves slightly upon the results of the first named author and Koldobsky which included a doubly-logarithmic error. The proof is based on an efficient way of discretizing the unit sphere.

arXiv comment16 pages. This is the published version, which has been modified in Section 4 to address an inaccuracy pointed out by Julian Haddad (in the previous version, Step 3 implicitly yielded an unnecessary log-star factor)