Resilience of cube slicing in ${\mathrm{\ell }}_p$

Ball’s celebrated cube slicing (1986) asserts that among hyperplane sections of the cube in ${\mathbb{R}}^n$, the central section orthogonal to $(1,1,0,\dots ,0)$ has the greatest volume. We show that the same continues to hold for slicing ${\mathrm{\ell }}_p$ balls when $p>{10}^{15}$, as well as that the same hyperplane minimizes the volume of projections of ${\mathrm{\ell }}_q$ balls for $1<q<1+{10}^{-12}$. This extends Szarek’s optimal Khinchin inequality (1976) which corresponds to $q=1$. These results thus address the resilience of the Ball–Szarek hyperplane in the ranges $2<p<\mathrm{\infty }$ and $1<q<2$, where analysis of the extremizers has been elusive since the works of Koldobsky (1998), Barthe–Naor (2002) and Oleszkiewicz (2003).