Diversities and the Generalized Circumradius

Abstract

The generalized circumradius of a set of points $A\subseteq {\mathbb{R}}^d$ with respect to a convex body $K$ equals the minimum value of $\lambda \ge 0$ such that $A$ is contained in a translate of $\lambda K$. 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 generalised circumradius to a finite subset of ${\mathbb{R}}^d$. We obtain elegant characterizations in the case that $K$ is a simplex or parallelotope.

To be published in Discrete and Computational Geometry