The problem of convexity of Chebyshev sets

Balaganskii, V. S.; Vlasov, L. P.

doi:10.1070/rm1996v051n06abeh003002

Cited by 68 publications

(45 citation statements)

References 8 publications

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“…Approximately compact sets have the property that any Chebyshev set which is approximately compact has to be convex, see [2]. In Section 8 a local version of this result is presented, see (11) and Theorem 8.7 below.…”

Section: Remark 32mentioning

confidence: 99%

“…By the Kuratowski-Zorn Lemma we find I ⊂ N such that j 1 = j 2 , whenever j 1 , j 2 (22) holds true for all x ∈ S, and by the L.P. Vlasov results the set S is convex, but this is impossible. Theorem 8.10 when compared with [2, Theorem 2.19] has the following differences: first, it is given in a Hilbert space, while [2,Theorem2.19] is given in a more general space, namely in the smooth Efimov-Stechkin space; second, it is not assumed that its boundary is included in a countable union of hyperplanes, as it was done in [2,Theorem]. So, it is natural to expect a result combining advantages of both theorems, but this is not the aim of this paper.…”

Section: If Int H \ (Int D S I 1 (S) ∪ Int D S I 2 (S) ∪ S) = ∅ Thenmentioning

confidence: 99%

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On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

Zagrodny

2015

Set-Valued Var. Anal

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The closure of preimages (inverse images) of metric projection mappings to a given set in a Hilbert space are investigated. In particular, some properties of fibers over singletons (level sets or preimages of singletons) of the metric projection are provided. One of them, a sufficient condition for the convergence of minimizing sequence for a giving point, ensures the convergence of a subsequence of minimizing points, thus the limit of the subsequence belongs to the image of the metric projection. Several examples preserving this sufficient condition are provided. It is also shown that the set of points for which the sufficient condition can be applied is dense in the boundary of the preimage of each set from a large class of subsets of the Hilbert space. As an application of obtained properties of preimages we show that if the complement of a nonconvex set is a countable union of preimages of convex closed sets then there is a point such that the value of the metric projection mapping is not a singleton. It is also shown that the Klee result, stating that only convex closed sets can be weakly closed Chebyshev sets, can be obtained for locally weakly closed sets. This paper is dedicated to Professor Lionel Thibault to honor great man and great mathematician. Thank You Lionel for the journeys in mathematics which we have gone together. Keywords

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Section: Remark 32mentioning

confidence: 99%

Section: If Int H \ (Int D S I 1 (S) ∪ Int D S I 2 (S) ∪ S) = ∅ Thenmentioning

confidence: 99%

On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

Zagrodny

2015

Set-Valued Var. Anal

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“…d C ðxÞ imply the existence of a convergent subsequence c n k and that the cardinality of the complement A c is less than the cardinality of continuum, then C is convex. For more results in a smooth space see the excellent survey paper of Balaganskii and Vlasov [4] and the references therein.…”

Section: Introductionmentioning

confidence: 98%

“…In a finite-dimensional Hilbert space Bunt [10], Kritikos [24], and Jessen [21] proved that every Chebyshev set is convex. However, in an infinitedimensional Hilbert space this problem is still open (see [1,4,15,22]). …”

Section: Introductionmentioning

confidence: 99%

A Chebyshev Set and its Distance Function

Wu¹

2002

Journal of Approximation Theory

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We prove that in a Banach space X with rotund dual X n a Chebyshev set C is convex iff the distance function d C is regular on X =C iff d C admits the strict and G# a ateaux derivatives on X =C which are determined by the subdifferential @jjx À % x xjj for each x 2 X =C and % x x 2 P C ðxÞ :If X is a reflexive Banach space with smooth and Kadec norm then C is convex iff it is weakly closed iff P C is continuous. If the norms of X and X n are Fr! e echet differentiable then C is convex iff d C is Fr! e echet differentiable on X =C: If also X has a uniformly G# a ateaux differentiable norm then C is convex iff the G# a ateaux (Fr! e echet) subdifferential @ À d C ðxÞ (@ F d C ðxÞ) is nonempty on X =C: # 2002 Elsevier Science (USA)

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“…whether every Chebyshev subset of a Hilbert space is convex. Surveys giving various partial answers of this problem were given by Vlasov (1973), Narang (1977), Deutsch (1993) and by Balaganski and Vlasov (1996). Metric projections have been very helpful in giving some partial answers of this problem.…”

Section: Introductionmentioning

confidence: 99%

A note on the convexity of Chebyshev sets

Narang

Sangeeta²

2009

B Soc Paran Mat

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Perhaps one of the major unsolved problem in Approximation Theory is : 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 metric spaces.

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The problem of convexity of Chebyshev sets

Cited by 68 publications

References 8 publications

On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

A Chebyshev Set and its Distance Function

A note on the convexity of Chebyshev sets

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