On property-$\bm{(R_1)}$ and relative Chebyshev centers in Banach spaces-II

We continue to study (strong) property-$(R_1)$ in Banach spaces. As discussed by Pai \& Nowroji in [{\it On restricted centers of sets}, J. Approx. Theory, {\bf 66}(2), 170--189 (1991)], this study corresponds to a triplet $(X,V,\mathcal{F})$, where $X$ is a Banach space, $V$ is a closed convex set, and $\mathcal{F}$ is a subfamily of closed, bounded subsets of $X$. It is observed that if $X$ is a Lindenstrauss space then $(X,B_X,\mathcal{K}(X))$ has strong property-$(R_1)$, where $\mathcal{K}(X)$ represents the compact subsets of $X$. It is established that for any $F\in\mathcal{K}(X)$, $\textrm{Cent}_{B_X}(F)\neq\emptyset$. 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 $\textrm{Cent}_{B_X}$ is Lipschitz continuous in $\mathcal{K}(X)$ if $X$ is a Lindenstrauss space. If $Y$ is a subspace of a Banach space $X$ and $\mathcal{F}$ 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-$(P_1)$) in $X$ for $F\in\mathcal{F}$ if $(X, Y, \mathcal{F}(X))$ satisfies strong property-$(R_1)$, where $\mathcal{F}(X)$ represents the set of all finite subsets of $X$. It is demonstrated that if $P$ is a bi-contractive projection in $\ell_\infty$, then $(\ell_\infty, Range (P), \mathcal{K}(\ell_\infty))$ exhibits the strong property-$(R_1)$, where $\mathcal{K}(\ell_\infty)$ represents the set of all compact subsets of $\ell_\infty$. Furthermore, stability results for these properties are derived in continuous function spaces, which are then studied for various sums in Banach spaces.