Resilience of cube slicing in ℓp

Source abstract

Ball’s celebrated cube slicing theorem asserts that among hyperplane sections of the cube in Rn, the central section orthogonal to (1,1,0,…,0) has the greatest volume. We show that the same continues to hold for slicing ℓp balls when p>1015, as well as that the same hyperplane minimizes the volume of projections of ℓq balls for 1<q<1+10−12. This extends Szarek’s optimal Khinchin inequality which corresponds to q=1. These results thus address the resilience of the Ball–Szarek hyperplane in the ranges 2<p<∞ and 1<q<2, where analysis of the extremizers has been elusive since the works of Koldobsky, Barthe and Naor, and Oleszkiewicz.