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arXiv · 2302.03000

A Note on the Chebyshev Set Problem in Normed Linear Spaces

Samson Owiti·Benard Okelo·Julia Owino
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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.

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Samson Owiti, Benard Okelo, Julia Owino. 2023-01-13. A Note on the Chebyshev Set Problem in Normed Linear Spaces. https://arxiv.org/abs/2302.03000

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