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 -compactness property (- F - SACP or SACP for short) for triplets (X, V, F), where X is a Banach space, V is a -closed subset of X, F is a subfamily of closed and bounded subsets of X, F is a collection of functions, and is the norm or weak topology on X. We characterize reflexive spaces with the Kadec–Klee property using triplets with - F - SACP. We investigate the relationship between - F - SACP and the continuity properties of the restricted f -center map. The study further examines - F - SACP in the context of CLUR -spaces and explores various characterizations of - F - SACP, including connections to reflexivity, Fréchet smoothness, and the Kadec–Klee property.
Das et al. (2026) studied this question.