Research Article

On property-(R ₁) and relative Chebyshev centers in Banach spaces-II

Published in: Quaestiones Mathematicae
Volume 47 , issue 2 , 2024 , pages: 461–476

DOI: 10.2989/16073606.2023.2229557

Author(s): Syamantak Das Indian Institute of Technology Hyderabad, India , Tanmoy Paul Indian Institute of Technology Hyderabad, India

Keywords: Primary: 41A28 , 41A65 , Secondary: 46B20 , 41A50 , Chebyshev center , restricted Chebyshev center , Property-(R ₁) , Lindenstrauss space , M-ideal

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Abstract

We continue to study (strong) property-(R ₁) in Banach spaces. As discussed by Pai & Nowroji in [On restricted centers of sets, J. Approx. Theory, 66(2), 170–189 (1991)], this study corresponds to a triplet , where X is a Banach space, V is a closed convex set, and is a subfamily of closed, bounded subsets of X. It is observed that if X is a Lindenstrauss space then has strong property-(R ₁), where represents the compact subsets of X. It is established that for any . This extends the well-known fact that a compact subset of a Lindenstrauss space X admits a nonempty Chebyshev center in X. We extend our observation that is Lipschitz continuous in if X is a Lindenstrauss space. If Y is a subspace of a Banach space X and represents the set of all finite subsets of B_X then we observe that B_Y exhibits the condition for simultaneously strongly proximinal (viz. property-(P ₁)) in X for if satisfies strong property-(R ₁), where represents the set of all finite subsets of X. It is demonstrated that if P is a bi-contractive projection in ℓ _∞, then exhibits the strong property-(R ₁), where represents the set of all compact subsets of ℓ _∞. Furthermore, stability results for these properties are derived in continuous function spaces, which are then studied for various sums in Banach spaces.

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