Resilience of cube slicing in $\ell_p$
Alexandros Eskenazis, Piotr Nayar, Tomasz Tkocz
TL;DR
This work establishes the resilience of Ball–Szarek hyperplane extremizers for volume-maximizing sections of $\ell_p^n$ balls and the dual Khinchin-type inequalities for projections of $\ell_q^n$ balls near the endpoint $q=1$. The authors develop probabilistic representations for section and projection volumes, combine stability results for Ball's and Szarek's inequalities with an inductive dimension-reduction framework, and obtain explicit constants enabling global results for large $p$ and small $q-1$. Their two-case approach (far from and near the extremizer) yields a full proof in the stated ranges, thereby extending the known extremizers beyond $p\le 2$ and $q\le 2$ and addressing long-standing questions about extremizers in the $2<p<\infty$ and $1<q<2$ regimes. The results advance our understanding of when the central hyperplane continues to govern extremal volume behavior in high dimensions, with precise quantitative stability that could inform related convex-geometry and functional-analytic problems.
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).