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arXiv · 2211.01986

Resilience of cube slicing in $\ell_p$

Alexandros Eskenazis·Piotr Nayar·Tomasz Tkocz
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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).

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Alexandros Eskenazis, Piotr Nayar, Tomasz Tkocz. 2022-11-03. Resilience of cube slicing in $\ell_p$. https://doi.org/10.1215/00127094-2024-0004

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