Abstract
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.
