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.
@article{9068,
title = {A note on the convexity of Chebyshev sets - doi: 10.5269/bspm.v27i1.9068},
journal = {Boletim da Sociedade Paranaense de Matem\'atica},
volume = {23},
year = {2009},
doi = {10.5269/bspm.v27i1.9068},
language = {EN},
url = {http://dml.mathdoc.fr/item/9068}
}
Narang, T. D.; Sangeeta, R. A note on the convexity of Chebyshev sets - doi: 10.5269/bspm.v27i1.9068. Boletim da Sociedade Paranaense de Matemática, Tome 23 (2009) . doi : 10.5269/bspm.v27i1.9068. http://gdmltest.u-ga.fr/item/9068/