Two characterizations of smooth norms
type: Item
instance of: scholarly article
title: Two characterizations of smooth norms
zbMATH Open document ID: 0952.52006
DOI: 10.1023/A:1005165709146
author: Horst Martini
author: Anita Schöbel
published in: Geometriae Dedicata
publication date: 2001-01-04T00:00:00Z
review text: Let \(\gamma\) be a norm with unit ball \(B = \{x\in\mathbb R^n: \gamma(x)\leqslant 1\}\), \(n\geqslant 2.\) The norm \(\gamma\) and its unit ball \(B\) are said to be smooth if for each \(t\in\mathbb R^n\) with \(\gamma(t) = 1\) there exists exactly one supporting hyperplane of \(B\) at \(t.\) The authors give the characterizations of smooth norms. The norm \(\gamma\) is smooth iff for all weighted point sets \(P\) not contained in a hyperplane each median hyperplane (with respect to \(\gamma\) and \(P\)) is the affine hull of \(n\) points from \(P,\) i.e., each median hyperplane satisfies the incidence criterion. The norm \(\gamma\) is smooth iff for all weighted point sets \(P\) not contained in a hyperplane each center hyperplane (with respect to \(\gamma\) and \(P\)) is at maximum distance to \(n + 1\) affinely independent points from \(P,\) i.e., every center hyperplane is blocked by \(P.\)
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 90B85
Mathematics Subject Classification ID: 52B55
zbMATH DE Number: 1347334
zbMATH Keywords: center hyperplane
zbMATH Keywords: centered hyperplane
zbMATH Keywords: distance
zbMATH Keywords: \(k\)-set problem
zbMATH Keywords: median hyperplane
zbMATH Keywords: Minkowski space
zbMATH Keywords: minsum hyperplane
zbMATH Keywords: norm
zbMATH Keywords: numerical approximation
zbMATH Keywords: point set width problem
zbMATH Keywords: robust statistics
zbMATH Keywords: smooth norm
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1347334