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

The maximal hyperplane section of the \(l_\infty ^n\)-ball, i.e. of the n-cube, is the one perpendicular to \(\frac{1}{\sqrt{2}} (1,1,0 ,\ldots ,0)\), 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. Suppose that \(p_0 \le p < \infty \). 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 \(20< p < p_0\).