A Study on the F-Simultaneous Approximative τ-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 F -simultaneous approximative τ -compactness property ( τ - F - SACP or SACP for short) for triplets (X, V,F) , where X is a Banach space, V is a τ -closed subset of X , F is a subfamily of closed and bounded subsets of X , F is a collection of functions, and τ is the norm or weak topology on X . We characterize reflexive spaces with the Kadec–Klee property using triplets with τ - F - SACP . We investigate the relationship between τ - F - SACP and the continuity properties of the restricted f -center map. The study further examines τ - F - SACP in the context of CLUR -spaces and explores various characterizations of τ - F - SACP , including connections to reflexivity, Fréchet smoothness, and the Kadec–Klee property.