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 (Figure presented.), where X is a Banach space, V is a closed convex set, and (Figure presented.) is a subfamily of closed, bounded subsets of X. It is observed that if X is a Lindenstrauss space then (Figure presented.) has strong property-(R 1), where (Figure presented.) represents the compact subsets of X. It is established that for any (Figure presented.). 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 (Figure presented.) is Lipschitz continuous in (Figure presented.) if X is a Lindenstrauss space. If Y is a subspace of a Banach space X and (Figure presented.) represents the set of all finite subsets of BX then we observe that BY exhibits the condition for simultaneously strongly proximinal (viz. property-(P 1)) in X for (Figure presented.) if (Figure presented.) satisfies strong property-(R 1), where (Figure presented.) represents the set of all finite subsets of X. It is demonstrated that if P is a bi-contractive projection in ℓ ∞, then (Figure presented.) exhibits the strong property-(R 1), where (Figure presented.) 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. © 2023 NISC (Pty) Ltd.