Resilience of cube slicing in $\ell_p$

TL;DR

This paper demonstrates that the cube slicing property extends to $\, ext{ell}_p$ balls for extremely large p and identifies the hyperplane minimizing projections of $\, ext{ell}_q$ balls for q near 1, addressing longstanding open problems.

Contribution

It extends the known cube slicing and projection minimization results to new ranges of p and q, revealing the resilience of these geometric properties.

Findings

Cube slicing property holds for $\, ext{ell}_p$ with p > 10^{15}.

Hyperplane minimizes projections of $\, ext{ell}_q$ balls for 1 < q < 1 + 10^{-12}.

Addresses open problems in the ranges 2 < p < ∞ and 1 < q < 2.

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).