A study on $\text{mr}F$-simultaneous approximative $\tau$-compactness property in Banach spaces

Veselý (1997) studied Banach spaces that admit $f$-centers for finite subsets of the space. In this work, we introduce the concept of $\text{mr}F$-simultaneous approximative $\tau$-compactness property ($\tau$-$\text{mr}F$-SACP in short) for triplets $(X,V,\text{mf}F)$, where $X$ is a Banach space, $V$ is a $\tau$-closed subset of $X$, $\text{mf}F$ is a subfamily of closed and bounded subsets of $X$, $\text{mr}F$ is a collection of functions, and $\tau$ is the norm or weak topology on $X$. We characterize reflexive spaces with the Kadec-Klee property using triplets with $\tau$-$\text{mr}F$-SACP. We investigate the relationship between $\tau$-$\text{mr}F$-SACP and the continuity properties of the restricted $f$-center map. The study further examines $\tau$-$\text{mr}F$-SACP in the context of $CLUR$ spaces and explores various characterizations of $\tau$-$\text{mr}F$-SACP, including connections to reflexivity, Fréchet smoothness, and the Kadec-Klee property.