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

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 $t\in \mathbb{R},$ $$\mu^+(K\cap(\xi^{\perp}+t\xi))\leq \frac{c}{\sqrt{n}}\mu(K)|K|^{-\frac{1}{n}}.$$ 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.