Diversities and the Generalized Circumradius

Abstract

The generalized circumradius of a set of points $A\subseteq R^d$ with respect to a convex body K equals the minimum value of $\lambda \ge 0$ such that a translate of $\lambda K$ contains A. Each choice of K gives a different function on the set of bounded subsets of $R^d$; 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.

Keywords: Convex geometry; Diversity; Generalized Minkowski spaces; Generalized circumradius; Metric geometry.