Uniform Hausdorff strong uniqueness
type: Item
instance of: scholarly article
title: Uniform Hausdorff strong uniqueness
zbMATH Open document ID: 0688.41040
DOI: 10.1016/0021-9045(89)90010-5
published in: Journal of Approximation Theory
publication date: 1989-01-01T00:00:00Z
review text: Let X be a normed linear space, M a proximinal subspace of X, \(P_ M(x)\) the set of all best approximations to x from M and \(P_ M\) a metric projection. In this article the definition of Hausdorff strong uniqueness of the set \(P_ M(x)\) and the definition of uniform Hausdorff strong uniqueness of the metric projection \(P_ M\) for any normed space X are given. The set \(P_ M(x)\) is said to be Hausdorff strongly unique if \[ r(x)=\inf \{\frac{\| x-m\| -d(x,M)}{d(m,P_ M(x))}:\quad m\in M\setminus P_ M(x)\}>0. \] The metric projection \(P_ M\) is said to be uniformly Hausdorff strongly unique if \(r(M)=\inf \{r(x):\) \(x\in X\}>0\). \textit{W. Li} [J. Approximation Theory 56, No.2, 164-184 (1989; Zbl 0672.41026)] defined and studied Hausdorff strong uniqueness in space C(T). In this article a characterization of (uniform) Hausdorff strong uniqueness is obtained. When M is a hyperplane, \(P_ M\) is Hausdorff strongly unique for some \(x\in X\setminus M\) if and only if \(P_ M\) is uniformly Hausdorff strongly unique. In C[a,b], \(P_ M\) is uniformly Hausdorff strongly unique if and only if M is one dimensional and Chebyshev.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 4127832
zbMATH Keywords: proximinal subspace
zbMATH Keywords: metric projection
zbMATH Keywords: Hausdorff strong uniqueness
zbMATH Keywords: uniform Hausdorff strong uniqueness
author: Sung Ho Park
reviewed by: D. Zarnadze
MaRDI profile type: Publication
full work available at URL: 0021-9045(89)90010-5
OpenAlex ID: W2084593986
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description: scientific article; zbMATH DE number 4127832