  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.
