Diversities and the Generalized Circumradius

The generalized circumradius of a set of points [Formula: see text] with respect to a convex body K equals the minimum value of [Formula: see text] such that a translate of [Formula: see text] contains A. Each choice of K gives a different function on the set of bounded subsets of [Formula: see text] ; we characterize which functions can arise in this way. Our characterization draws on the theory of diversities, a recently introduced generalization of metrics from functions on pairs to functions on finite subsets. We additionally investigate functions which arise by restricting the generalized circumradius to a finite subset of [Formula: see text] . We obtain elegant characterizations in the case that K is a simplex or parallelotope.