The generalised (uniform) Mazur Intersection Property

Abstract

Given a family C of closed bounded convex sets in a Banach space X, we say that X has the C-MIP if every C∈C is the intersection of closed balls containing it. In this paper, we introduce a stronger version of C-MIP and show that it is a more satisfactory generalisation of the MIP inasmuch as one can obtain complete analogues of various characterisations of the MIP. We also introduce uniform versions of the (strong) C-MIP and characterise them analogously. Even in this case, the strong C-UMIP appears to have richer characterisations than the C-UMIP.