The dimension-free threshold for maximal sections of ${\ell }_p$

Abstract

We prove the existence of a sharp threshold $p_0\approx 26.265\dots$, such that for every $n\in \mathbb{N}$ and $p\ge p_0$, the volume maximizing section of the unit ball of ${\ell }_p^n$ is the section with Ball's hyperplane $(\frac{e_1+e_2}{\sqrt{2}}{)}^{\perp }$, whereas for $p\in (2,p_0)$ the asymptotic maximizer as the ambient dimension $n\to \infty$ is the hyperplane orthogonal to the diagonal vector $\frac{e_1+\cdots +e_n}{\sqrt{n}}$.