Diversities and the Generalized Circumradius

Abstract

The generalized circumradius of a set of points Asubseteq mathbb {R}^d with respect to a convex body K equals the minimum value of lambda ge 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 mathbb {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 mathbb {R}^d. We obtain elegant characterizations in the case that K is a simplex or parallelotope.