### Ball hulls, ball intersections, and 2-center problems for gauges

#### Abstract

The notions of ball hull and ball intersection of nite sets, important in Banach space theory, are extended from normed planes to generalized normed planes, i.e., to (asymmetric) convex distance functions which are also called gauges. In this more general setting we derive various new results about these notions and their relations to each other. Further on, we extend the known 2-center problem and a modified version of it from the Euclidean situation to norms and gauges or, in other words, from Euclidean circles to arbitrary closed convex curves. We derive algorithmical results on the construction of ball hulls and ball intersections, and computational approaches to the 2-center problem with constrained circles and, in case of strictly convex norms and gauges, for the fixed 2-center problem are also given.

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