On hyperplane sections and projections in $l_p^n$

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