A Study on the $$\mathscr{F}$$-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 the $$\mathscr{F}$$ -simultaneous approximative $$\tau$$ -compactness property ( $$\tau$$ - $$\mathscr{F}$$ - $$\mathrm{SACP}$$ or SACP for short) for triplets $$(X, V,\mathfrak{F})$$ , where $$X$$ is a Banach space, $$V$$ is a $$\tau$$ -closed subset of $$X$$ , $$\mathfrak{F}$$ is a subfamily of closed and bounded subsets of $$X$$ , $$\mathscr{F}$$ 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 $$\tau$$ - $$\mathscr{F}$$ - $$\mathrm{SACP}$$ . We investigate the relationship between $$\tau$$ - $$\mathscr{F}$$ - $$\mathrm{SACP}$$ and the continuity properties of the restricted $$f$$ -center map. The study further examines $$\tau$$ - $$\mathscr{F}$$ - $$\mathrm{SACP}$$ in the context of $$\mathrm{CLUR}$$ -spaces and explores various characterizations of $$\tau$$ - $$\mathscr{F}$$ - $$\mathrm{SACP}$$ , including connections to reflexivity, Fréchet smoothness, and the Kadec–Klee property.