Maximal hyperplane sections of the unit ball of $l_p$ for $p>2$

The maximal hyperplane section of the $l_\infty^n$-ball, i.e. of the $n$-cube, is the one perpendicular to 1/sqrt 2 (1,1,0, ... ,0), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the $l_p^n$-balls for very large $p \ge 10^{15}$. We show that the analogue of Ball's result essentially holds for all $26.265 \simeq p_0 \le p < \infty$. By Oleszkiewicz, Ball's result does not transfer to $l_p^n$ for $2 < p < p_0$. Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions $n$. In this case, we give an asymptotic upper bound for $20 < p < p_0$.