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

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 $$\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left(\frac{{\sin }^{2}t}{t^2}\right)}^{p}dt\le \frac{\sqrt{2}}{\sqrt{p}},p\ge 1,$$ 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.