A Note on the Chebyshev Set Problem in Normed Linear Spaces

TL;DR

This paper investigates the convexity of Chebyshev sets in normed linear spaces, providing characterizations and conditions under which these sets are convex, especially in the context of real-valued norm-attainable functions.

Contribution

It characterizes Chebyshev sets and their convexity in normed linear spaces, particularly for real-valued norm-attainable functions under specific smoothness and topological conditions.

Findings

Chebyshev sets are convex if they are closed, rotund, and admit Gateaux and Fréchet differentiability.

The paper provides a characterization of Chebyshev sets in general normed linear spaces.

Convexity of Chebyshev sets is established in the space of real-valued norm-attainable functions under certain conditions.

Abstract

Best approximation (BA) 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 (CS) which is a subset of a normed linear space (NLS) which contains unique BAs. However, a fundamental question remains unsolved to-date regarding the convexity of the CS in infinite NLS known as the CS problem. The question which has not been answered is: Is every CS in a NLS convex?. This question has not got any solution including the simplest form of a real Hilbert space (HS). In this note, we characterize CSs and convexity in NLSs. In particular, we consider the space of all real-valued norm-attainable functions. We show that CSs of the space of all real-valued norm-attainable functions are convex when they are closed, rotund and admits both Gateaux and Fr\'{e}chet differentiability conditions.