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

TL;DR

This paper establishes an optimal lower bound for Koldobsky's slicing inequality using a novel discretization approach, improving previous results by reducing the error term.

Contribution

It introduces a new method of discretizing the unit sphere to derive an optimal lower bound for the slicing inequality, refining prior bounds.

Findings

The bound is optimal up to a universal constant.

The method improves previous results by reducing the error term.

The approach involves an efficient discretization of the sphere.

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