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

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