Quantitative form of Ball's Cube slicing in $\mathbb{R}^n$ and equality cases in the min-entropy power inequality

TL;DR

This paper establishes a quantitative version of Ball's cube slicing theorem and derives equality cases in the min-entropy power inequality, also providing a specialized quantitative form of Khintchine's inequality for p=1.

Contribution

It introduces a quantitative form of Ball's cube slicing theorem and applies it to identify equality cases in the min-entropy power inequality, along with a new quantitative version of Khintchine's inequality for p=1.

Findings

Quantitative bounds for Ball's cube slicing theorem.

Identification of equality cases in the min-entropy power inequality.

A new quantitative form of Khintchine's inequality for p=1.

Abstract

We prove a quantitative form of the celebrated Ball's theorem on cube slicing in $\mathbb{R}^n$ and obtain, as a consequence, equality cases in the min-entropy power inequality. Independently, we also give a quantitative form of Khintchine's inequality in the special case $p=1$.