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.

We prove that in a convex metric space (X,d), an existence set K having a lower semi continuous metric projection is a ?-sun and in a complete M-space, a Chebyshev set K with a continuous metric projection is a ?-sun as well as almost convex.