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

Abstract

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 ${\ell }_p$ balls when $p>10^{15}$, as well as that the same hyperplane minimizes the volume of projections of ${\ell }_q$ 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 $2<p<\infty$ and $1<q<2$, where analysis of the extremizers has been elusive since the works of Koldobsky (1998), Barthe--Naor (2002) and Oleszkiewicz (2003).

Final version. To appear in Duke Math. J

References in corpus (3)

• Slicing ${\ell }_p$-balls reloaded: stability, planar sections in ${\ell }_1$
• Extremal sections and projections of certain convex bodies: a survey
• Distributional stability of the Szarek and Ball inequalities