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A Note on The Convexity of Chebyshev Sets

Author(s): T.D. Narang | Sangeeta

Journal: Boletim da Sociedade Paranaense de Matemática
ISSN 0037-8712

Volume: 27;
Issue: 1;
Start page: 59;
Date: 2009;
Original page

Keywords: Convex set | Cheybshev set | Convex space | Strongly convex space | Metric projection | Non-expansive map.

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.
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