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 $$\frac{1}{\pi}\int_{-\infty}^{\infty} \left(\frac{\sin^2 t}{t^2}\right)^pdt\leq \frac{\sqrt 2}{\sqrt p}, \quad p\geq 1,$$ 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.