Diversities and the Generalized Circumradius.

The of a set of points with respect to a convex body equals the minimum value of such that a translate of contains . Each choice of gives a different function on the set of bounded subsets of ; we characterize which functions can arise in this way. Our characterization draws on the theory of , 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 . We obtain elegant characterizations in the case that is a simplex or parallelotope.