On hyperplane sections and projections in \(l_p^n\)

For \(2< p < p_0 \simeq 26.265\) $2<p<p_0\simeq 26.265$ , the hyperplane section of the \(l_p^n\) $l_p^n$ -unit ball \(B_p^n\) $B_p^n$ perpendicular to \(a^{(n)} = \frac{1}{\sqrt{n}} (1 ,\ldots ,1)\) $a^{(n)}=\frac{1}{\sqrt{n}}(1,\dots ,1)$ for large n has larger volume than the one orthogonal to \(a^{(2)} = \frac{1}{\sqrt{2}} (1,1,0 ,\ldots ,0)\) $a^{(2)}=\frac{1}{\sqrt{2}}(1,1,0,\dots ,0)$ , as shown by Oleszkiewicz. This is different from the case of \(l_\infty ^n\) $l_{\infty }^n$ considered by Ball. We give a quantitative estimate for which dimensions n this happens, namely for \(n > c (\frac{1}{p_0-p} + \frac{1}{p-2})\) $n>c(\frac{1}{p_0-p}+\frac{1}{p-2})$ for some absolute constant \(c>0\) $c>0$ . Correspondingly for projections of \(B_q^n\) $B_q^n$ onto hyperplanes, Barthe and Naor showed that projections onto hyperplanes perpendicular to \(a^{(n)}\) $a^{(n)}$ have smaller volume for large n than onto the one orthogonal to \(a^{(2)}\) $a^{(2)}$ , if \(\frac{4}{3}< q < 2\) $\frac{4}{3}<q<2$ , different from the case \(q=1\) $q=1$ . We show that this happens for all \(n > 5 (\frac{1}{q-\frac{4}{3}} + \frac{1}{2-q})\) $n>5(\frac{1}{q-\frac{4}{3}}+\frac{1}{2-q})$ .