On minimizing sequences for \(k\)-centres
type: Item
instance of: scholarly article
title: On minimizing sequences for \(k\)-centres
zbMATH Open document ID: 1013.60002
DOI: 10.1016/S0021-9045(02)00010-2
author: J. Lember
published in: Journal of Approximation Theory
publication date: 2003-04-02T00:00:00Z
review text: Let \(P\) be a Borel measure on a separable metric space \((E,d)\), \(k\geq 1\), and \(\varphi\) be an increasing function. The author considers the problem of approximating \(P\) by \(k\)-element subsets \(A\subset E\) by minimization of the function \(W(A)=\int \varphi (d(x,A))P(dx)\). For example, if \(E\) is a separable dual Banach space with the property: \(x_n\rightarrow\) weakly*\(x\) and \(\|x_n\|\rightarrow \|x\|\) imply \(\|x_n-x\|\rightarrow 0\), then every minimizing sequence \(A_n\) has a norm convergence subsequence. The problem arises from the investigation of consistency of empirical \(k\)-centres in mathematical statistics. The paper is based on the author's doctoral dissertation.
Mathematics Subject Classification ID: 60B99
Mathematics Subject Classification ID: 46N30
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 1891274
zbMATH Keywords: \(k\)-centre
zbMATH Keywords: empirical measure
zbMATH Keywords: Kadec-Klee property
reviewed by: A. M. Plichko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1891274