A Note on The Convexity of Chebyshev Sets

Perhaps one of the major unsolved problem in Approximation Theoryis: Whether or not every Chebyshev subset of a Hilbert space must be convex. Many partial answers to this problem are available in the literature. R.R. Phelps[Proc. Amer. Math. Soc. 8 (1957), 790-797] showed that a Chebyshev set in an...

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Main Authors:	Sangeeta, T.D. Narang
Format:	Article
Language:	English
Published:	Sociedade Brasileira de Matemática 2009-07-01
Series:	Boletim da Sociedade Paranaense de Matemática
Subjects:	Convex set Cheybshev set Convex space Strongly convex space Metric projection Non-expansive map.
Online Access:	http://www.periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/9068/5272

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spelling	doaj-083b275965bc4dc0bbcbc34997dbe3e82020-11-24T21:29:57ZengSociedade Brasileira de MatemáticaBoletim da Sociedade Paranaense de Matemática0037-87122175-11882009-07-012715963A Note on The Convexity of Chebyshev SetsSangeetaT.D. NarangPerhaps one of the major unsolved problem in Approximation Theoryis: Whether or not every Chebyshev subset of a Hilbert space must be convex. Many partial answers to this problem are available in the literature. R.R. Phelps[Proc. Amer. Math. Soc. 8 (1957), 790-797] showed that a Chebyshev set in an inner product space (or in a strictly convex normed linear space) is convex if the associated metric projection is non-expansive. We extend this result to metricspaces.http://www.periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/9068/5272Convex setCheybshev setConvex spaceStrongly convex spaceMetric projectionNon-expansive map.
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Sangeeta T.D. Narang
spellingShingle	Sangeeta T.D. Narang A Note on The Convexity of Chebyshev Sets Boletim da Sociedade Paranaense de Matemática Convex set Cheybshev set Convex space Strongly convex space Metric projection Non-expansive map.
author_facet	Sangeeta T.D. Narang
author_sort	Sangeeta
title	A Note on The Convexity of Chebyshev Sets
title_short	A Note on The Convexity of Chebyshev Sets
title_full	A Note on The Convexity of Chebyshev Sets
title_fullStr	A Note on The Convexity of Chebyshev Sets
title_full_unstemmed	A Note on The Convexity of Chebyshev Sets
title_sort	note on the convexity of chebyshev sets
publisher	Sociedade Brasileira de Matemática
series	Boletim da Sociedade Paranaense de Matemática
issn	0037-8712 2175-1188
publishDate	2009-07-01
description	Perhaps one of the major unsolved problem in Approximation Theoryis: Whether or not every Chebyshev subset of a Hilbert space must be convex. Many partial answers to this problem are available in the literature. R.R. Phelps[Proc. Amer. Math. Soc. 8 (1957), 790-797] showed that a Chebyshev set in an inner product space (or in a strictly convex normed linear space) is convex if the associated metric projection is non-expansive. We extend this result to metricspaces.
topic	Convex set Cheybshev set Convex space Strongly convex space Metric projection Non-expansive map.
url	http://www.periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/9068/5272
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A Note on The Convexity of Chebyshev Sets

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