Continuity Properties of the Ball Hull Mapping

The ball hull mapping β associates with each closed bounded convex setK in a Banach space its ball hull β(K), defined as the intersection of all closed balls containing K. We are concerned in this paper with continuity and Lipschitz continuity (with respect to the Hausdorff metric) of the ball hull mapping. It is proved that β is a Lipschitz map in finite dimensional polyhedral spaces. Both properties, finite dimension and polyhedral norm, are necessary for this result. Characterizing the ball hull mapping by means of H-convexity we show, with the help of a remarkable example from combinatorial geometry, that there exist norms with noncontinuous β map, even in finite dimensional spaces. Using this surprising result, we then show that there are infinite dimensional polyhedral spaces (in the usual sense of Klee) for which the map β is not continuous. A property known as ball stability implies that β has Lipschitz constant one. We prove that every Banach space of dimension greater than two can be renormed so that there is an intersection of closed balls for which none of its parallel bodies is an intersection of closed balls, thus lacking ball stability.