Diversities and the Generalized Circumradius

Abstract

AbstractThe generalized circumradius of a set of points $$A\subseteq \mathbb {R}^d$$ A ⊆ R d with respect to a convex body K equals the minimum value of $$\lambda \ge 0$$ λ ≥ 0 such that a translate of $$\lambda K$$ λ K contains A. Each choice of K gives a different function on the set of bounded subsets of $$\mathbb {R}^d$$ 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 $$\mathbb {R}^d$$ R d . We obtain elegant characterizations in the case that K is a simplex or parallelotope.