Diversities and the generalized circumradius

The generalized circumradius of a set of points A\subseteq R^d with respect to a convex body K equals the minimum value of \lambda\geq 0 such that a translate of  \lambda 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.