Slicing \(\ell_p\)-balls reloaded: stability, planar sections in \(\ell_1\)
type: Item
instance of: scholarly article
title: Slicing \(\ell_p\)-balls reloaded: stability, planar sections in \(\ell_1\)
zbMATH Open document ID: 1506.52003
DOI: 10.1214/22-AOP1584
author: Giorgos Chasapis
author: Piotr Nayar
author: Tomasz Tkocz
published in: The Annals of Probability
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2109.05645
review text: Let \(n\) be an integer; for a real number \(p>0\), let \[ B^n_p=\left\{x=(x_1,\dots,x_n)\in\mathbb{R}^n:\sum_{i=1}^n|x_i|^p\leq1\right\}. \] be the unit \(\ell_p\)-ball in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\). In particular, for \(p=1\), \(B^n_1\) is the so-called \textit{cross-polytope}. Let \(k\in\{1,\dots,n\}\) and \(H\) be an hyperplane of \(\mathbb{R}^n\) of dimension \(k\). The \(k\)-dimensional Hausdorff measure of the intersection of \(B^n_p\) with \(H\) is denoted by \(\operatorname{vol}_k(B^n_p\cap H)\). The problem of finding the maximum and minimum values of \(\operatorname{vol}_k(B^n_p\cap H)\), as \(H\) ranges over all possible \(k\)-dimensional hyperplanes, for various values of \(k\) and \(p\), is well studied. New results are provided by the authors in this paper. The first result is a sharp lower bound when \(p=1\) and \(k=2\). Theorem. Let \(n\geq 3\); for every \(2\)-dimensional hyperplane \(H\) \[ \operatorname{vol}_2(B^n_1\cap H)\geq \frac{n^2\sin^3\left(\frac{\pi}{2n}\right)}{\cos\left(\frac\pi{2n}\right)}. \] If equality holds in the previous inequality, then \(B_1^n\cap H\) is isometric to a regular planar \(2n\)-gon. The minimum is achieved for \(H=T(\mathbb{R}^2)\), where \(T:\mathbb{R}^2\to\mathbb{R}^n\) is defined by \(T(x)=(\langle x,v_1\rangle,\dots,\langle x,v_n\rangle)\), and \(v_k=\left(\cos\left(\frac{k\pi}{n}\right),\sin\left(\frac{k\pi}{n}\right)\right)\), for \(k=1,\dots,n\). The minimizing subspace is unique up to permutations of the coordinates, and reflections. The second main contribution of the paper is a stability result for the known cases of subspaces minimizing or maximizing the volume of their intersection with \(B^n_p\). Theorem. Let \(n\geq 1\). There exists a positive constant depending on \(p\), \(c_p\), such that for every unit vector \(a=(a_1,\dots,a_n)\), with \(a_1\geq a_2\geq \dots\geq a_n\), the following inequalities hold \begin{align*} \frac{\operatorname{vol}_{n-1}(B_p^n\cap a^\perp)}{\o…
reviewed by: A. Colesanti
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 7608253
zbMATH Keywords: convex bodies
zbMATH Keywords: volume of sections
zbMATH Keywords: stability
zbMATH Keywords: \(p\)-norm
zbMATH Keywords: negative moments
MaRDI profile type: Publication
arXiv ID: 2109.05645
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description: scientific article