An asymptotically sharp form of Ball's inequality by probability methods

TL;DR

This paper improves Ball's inequality using probability methods, providing a sharper asymptotic form that better captures the decay rate and equality conditions, with applications to geometric volume bounds.

Contribution

The paper introduces an asymptotically sharp version of Ball's inequality by employing probabilistic techniques, refining the decay rate and equality characterization.

Findings

Derived an improved form of Ball's inequality with sharper asymptotics

Established the correct decay rate aligning with the limit ratio

Applied the inequality to bound volumes of cube sections

Abstract

To prove by probabilistic methods that every $(n-1)$-dimensional section of the unit cube in $R^n$ has volume at most $\sqrt 2$, K. Ball made essential use of the inequality $$ \frac{1}{\pi}\int_{-\infty}^{\infty} \left(\frac{\sin^2 t}{t^2}\right)^pdt\leq \frac{\sqrt 2}{\sqrt p}, \quad p\geq 1, $$ in which equality holds if and only if $p=1$. The right side of above inequality has the correct rate of decay though the limit of the ratio of the right to left side is ${\sqrt{\frac{3}{\pi}}}$ rather then $\sqrt 2$. Applying Ball's methods we put all of this into the improved form of the Ball's inequality.