Successive radii and ball operators in generalized Minkowski spaces

We introduce successive radii in generalized Minkowski spaces (that is, with respect to gauges) and study some first properties. This is done via formulating some kind of minimal containment problems, where intersections or Minkowski sums of the latter set and affine flats of a certain dimension are incorporated. Since this is strongly related to minimax location problems, we also look at ball intersections and ball hulls.