Extremal Radii, Diameter, and Minimum Width in Generalized Minkowski Spaces

We discuss the notions of circumradius, inradius, diameter, and minimum width in generalized Minkowski spaces (that is, with respect to gauges), i.e., we measure the "size" of a given convex set in a finite-dimensional real vector space with respect to another convex set. This is done via formulating some kind of containment problem incorporating homothetic bodies of the latter set or strips bounded by parallel supporting hyperplanes thereof. The paper can be seen as a theoretical starting point for studying metrical problems of sets in generalized Minkowski spaces.