On convexity of chebyshev sets in norm-attainable classes

Abstract

Best approximation is an interesting field in functional analysis that has attracted a lot of attention from many researchers for a very long period of time up-to-date. Of greatest consideration is the characterization of the Chebyshev set which is a subset of a normed linear space which contains unique best approximations. However, a fundamental question remains unsolved to-date regarding the convexity of the Chebyshev set in infinite normed linear space known as the Chebyshev set problem. The question which has not been answered is: Is every Chebyshev set in a normed linear space convex? This question has not got any solution including the simplest form of a real Hilbert space. The main objective of this study is to characterize Chebyshev sets and convexity in normed linear spaces. The specific objectives include to: Determine conditions under which subsets of normed linear spaces are Chebyshev; Characterize distance functions of Chebyshev set in normed linear spaces and; Investigate convexity of Chebyshev set in normed linear spaces. The methodology involved the use of known fundamental principles like the Bunt-Motzkin Theorem, Fre`chet and Gateaux differentiability conditions and best approximation techniques. The results of this study show that every non-void closed set in N A(H ) is a Chebyshev set if it is Euclidean. Also, if U is proximal in N A(H ) then U is closed and a Chebyshev set. Every distance function of a Chebyshev set of the normed linear space of all norm-attainable real- valued functions is Fr´echet differentiable. For convexity, it has been shown that every PU (ξ) which is convex is a Chebyshev set. The results of this study have provided a partial solution to the Chebyshev set problem and are also useful in solving convex optimization problems.