On property-(R1) and relative Chebyshev centers In Banach spaces-II

Abstract

We continue to study (strong) property-(R₁) 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 (X; V;F), where X is a Banach space, V is a closed convex set, and F is a subfamily of closed, bounded subsets of X. It is observed that if X is a Lindenstrauss space then (X;B_X;K(X)) has strong property-(R₁), where K(X) represents the compact subsets of X. It is established that for any F ∈ K(X), Cent_BX(F) ̸= ∅. 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 Cent_BX is Lipschitz continuous in K(X) if X is a Lindenstrauss space. If Y is a subspace of a Banach space X and F 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-(P₁)) in X for F ∈ F if (X; Y;F(X)) satisfies strong property-(R₁), where F(X) represents the set of all infite subsets of X. It is demonstrated that if P is a bi-contractive projection in ℓ∞ , then (ℓ∞;Range(P);K(ℓ∞)) exhibits the strong property-(R1), where K(ℓ∞) 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.