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

The maximal hyperplane section of the $l_{\mathrm{\infty }}^n$-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 $p\ge {10}^{15}$. By Oleszkiewicz, Ball’s result does not transfer to $l_p^n$ for $2<p<p_0\simeq 26.265$. 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 $l_p^n$-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 $\ge 2^{-p}$ to the even integers. Under similar assumptions, we give a Gaussian upper bound for $20<p<p_0$.