A Note on The Convexity of Chebyshev Sets

Abstract

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.

Keywords

• Convex set
• Cheybshev set
• Convex space
• Strongly convex space
• Metric projection
• Non-expansive map.