On maximal hyperplane sections of the unit ball of \(l_p^n\) for \(p>2\)

The maximal hyperplane section of the \(l_\infty ^n\) $l_{\infty }^n$ -ball, i.e. of the n-cube, is the one perpendicular to \(\frac{1}{\sqrt{2}} (1,1,0 ,\ldots ,0)\) $\frac{1}{\sqrt{2}}(1,1,0,\dots ,0)$ , as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the \(l_p^n\) $l_p^n$ -balls for very large \(p \ge 10^{15}\) $p\ge {10}^{15}$ . By Oleszkiewicz, Ball’s result does not transfer to \(l_p^n\) $l_p^n$ for \(2< p < p_0 \simeq 26.265\) $2<p<p_0\simeq 26.265$ . Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions n. Suppose that \(p_0 \le p < \infty \) $p_0\le p<\infty$ . We show that the analogue of Ball’s result holds in \(l_p^n\) $l_p^n$ -balls for all hyperplanes with normal unit vectors a, if all coordinates of a have modulus \(\le \frac{1}{\sqrt{2}}\) $\le \frac{1}{\sqrt{2}}$ and p has distance \(\ge 2^{-p}\) $\ge 2^{-p}$ to the even integers. Under similar assumptions, we give a Gaussian upper bound for \(20< p < p_0\) $20<p<p_0$ .