Intersections of Balls and the Ball Hull Mapping

We are concerned with some properties of the family of all subsets of a Banach space that can be written as an intersection of balls. A space with the Mazur Intersection Property (MIP) always satisfies those properties, so they can be regarded as weakenings of the MIP in different directions. The "ball hull" function (mapping a set to the intersection of all closed balls that cover it) is often an effective tool to study those properties.