The dimension-free threshold for maximal sections of $\ell_p$

Abstract

We prove the existence of a sharp threshold $p_0 \approx 26.265\ldots$, such that for every $n\in\mathbb{N}$ and $p\geq p_0$, 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 $\frac{e_1+\cdots+e_n}{\sqrt{n}}$.