On suns and cosuns in finite dimensional normed real vector spaces
type: Item
instance of: scholarly article
title: On suns and cosuns in finite dimensional normed real vector spaces
zbMATH Open document ID: 0592.41043
DOI: 10.1007/BF01955023
author: L. Hetzelt
published in: Acta Mathematica Hungarica
publication date: 1985-01-01T00:00:00Z
review text: Given a finite-dimensional normed linear space X and a nonempty subset A of X, we define \(\bar x\in A\) to be a best coapproximation to \(x\in X\) if \(\| \bar x-x\| =\min \{\| a-x\|;a\in A\}\). If we borrow from the theory of best approximation the notions of sun, Chebyshev set and so on, we can define the akin notions of cosun, co-Chebyshev set, and so on. Here a detailed study of the following notions is accomplished: sets defined by \(E_ X(x,y)=\{z\in X;\| x-z\| <\| y-z\| \}\) in X (especially in \(\ell^ p_ 3)\); cosuns and related notions; suns and cosuns in two-dimensional spaces, by using dual norms. This deep study is the author's doctoral thesis; note that while two- dimensional spaces are quite special, three-dimensional spaces already contain practically all the pathologies of general normed spaces. For some related problems, see the survey paper by the reviewer [Rend. Semin. Mat. Fis. Milano 53, 131-148 (1983)].
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 46B99
zbMATH DE Number: 3951228
zbMATH Keywords: best coapproximation
zbMATH Keywords: cosun
zbMATH Keywords: co-Chebyshev set
reviewed by: Pier Luigi Papini
MaRDI profile type: Publication
cites work: Contractive projections in \(L_ p\)-spaces
cites work: Points minimaux et ensembles optimaux dans les espaces de Banach
cites work: Q4153188
cites work: Über die beste Approximation im \({\mathbb{R}}^ n\)
cites work: Nonexpansive projections on subsets of Banach spaces
cites work: Q4163944
cites work: The foundations of Minkowskian geometry
cites work: Classes of Semi-Inner-Product Spaces
cites work: Über kennzeichnende Eigenschaften von Ellipsoiden und euklidischen Räumen. III
cites work: Fixpunktmengen von Kontraktionen in endlichdimensionalen normierten Räumen
cites work: Q3252895
cites work: Q5678003
cites work: Über die Erzeugung von Darstellungen von Gruppen durch Darstellungen von Untergruppen
cites work: Continuous selections. I
cites work: Best coapproximation in normed linear spaces
cites work: APPROXIMATIVE PROPERTIES OF SETS IN NORMED LINEAR SPACES
full work available at URL: https://doi.org/10.1007/bf01955023
OpenAlex ID: W2061521814
description: scientific article; zbMATH DE number 3951228