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

Abstract: The maximal hyperplane section of the $l_\infty<sup>n$-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 $l_p<sup>n$-balls for very large $p \ge 10<sup>{15}$. By Oleszkiewicz, Ball's result does not transfer to $l_p<sup>n$ for $2 &lt; p &lt; p_0 \simeq 26.265$. Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions $n$. We show that the analogue of Ball's result holds in $l_p<sup>n$-balls for all hyperplanes with normal unit vectors $a$, if all coordinates of $a$ have modulus $\le \frac 1 {\sqrt 2}$ and $p$ has distance $\ge 2<sup>{-p}$ to the even integers. Under similar assumptions, we give a Gaussian upper bound for $20 &lt; p &lt; p_0$.