On δ-suns

Abstract

We prove that an approximatively compact Chebyshev set in an M-space is a ?-sun and a ?-sun in a complete strong M-space (or externally convex M-space) is almost convex.

A note on suns in convex metric spaces

“…Since in a complete -space, every -sun is a -sun [9], and is almost convex [8], we have: Corollary 3.2. In a complete -space, a Chebyshev set with a continuous metric projection is a -sun and almost convex.…”

“…While making an attempt in this direction, Effimov and Steckin [4] introduced the concept of a sun and Vlasov [13] introduced the concepts of -, -, -, -suns and almost convex sets in Banach spaces. These concepts were extended to convex metric spaces in [8] and some of the results proved by Vlasov [13] in Banach spaces were also proved in convex metric spaces. Continuing the study taken up in [8] and [9], we prove that in a convex metric space ( , ), an existence set having a lower semi continuous metric projection is a -sun and in a complete -space, a Chebyshev set with a continuous metric projection is a -sun as well as almost convex.…”