Hyperplane projections of the unit ball of \(\ell_{p}^{n}\)
type: Item
instance of: scholarly article
title: Hyperplane projections of the unit ball of \(\ell_{p}^{n}\)
zbMATH Open document ID: 0999.52003
author: Assaf Naor
author: Franck Barthe
published in: Discrete \& Computational Geometry
publication date: 2002-05-30T00:00:00Z
review text: Inspired by various results on hyperplane sections of special unit, balls of normed linear spaces, the authors study extreme values of the volume of orthogonal hyperplane projections of unit balls of \(\ell^n_p\), i.e. of the form \[ B^n_p=\{x\in R^n:\sum^n_{i=1}|x_i|^p\leq 1\}\quad\text{with }1\leq p <\infty. \] They succeed in bringing the knowledge on this roughly to the same level as it is for sections, but they do not use the duality method. Also they show the following: If \(1\leq p\leq q =+\infty\) and \(H\) is a hyperplane of \(\mathbb{R}^n\), then \[ \frac{\text{vol}[\pi_H(B^n_q)]}{\text{vol}(B^{n-1}_q)}\geq \frac{\text{vol}[\pi_H(B^n_p)]}{\text{vol}(B^{n-1}_p)} , \] where \(\pi_H\) denotes the orthogonal projection onto \(H\).
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 1747938
zbMATH Keywords: Khinchine formula
zbMATH Keywords: Meyer-Pajor theorem
zbMATH Keywords: Busemann-Petty problem
zbMATH Keywords: hyperplane sections
zbMATH Keywords: unit balls of \(\ell^n_p\)
MaRDI profile type: Publication
full work available at URL: s00454-001-0066-3
OpenAlex ID: W2095075176
DOI: 10.1007/S00454-001-0066-3
description: scientific article; zbMATH DE number 1747938