Diversities and the Generalized Circumradius

The generalized circumradius of a set of points A ⊆ R d with respect to a convex body K equals the minimum value of λ ≥ 0 such that a translate of λ K contains A . Each choice of K gives a different function on the set of bounded subsets of R d ; 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 R d . We obtain elegant characterizations in the case that K is a simplex or parallelotope.