Slicing $\ell_p$-balls reloaded: stability, planar sections in $\ell_1$

TL;DR

This paper investigates the geometric properties of $\, ext{l}_p$-balls, revealing the extremal planar sections and establishing stability results through probabilistic methods, enhancing understanding of convex body sections.

Contribution

It provides new stability results for hyperplane sections of $\, ext{l}_p$-balls and characterizes the minimal-volume sections of cross-polytopes, using probabilistic techniques.

Findings

The minimal-volume central section of the $n$-dimensional cross-polytope is a regular $2n$-gon.

Stability results are established for hyperplane sections of $\, ext{l}_p$-balls.

Probabilistic methods link negative moments of projections to section volumes.

Abstract

We show that the two-dimensional minimum-volume central section of the $n$-dimensional cross-polytope is attained by the regular $2n$-gon. We establish stability-type results for hyperplane sections of $\ell_p$-balls in all the cases where the extremisers are known. Our methods are mainly probabilistic, exploring connections between negative moments of projections of random vectors uniformly distributed on convex bodies and volume of their sections.