Diversities and the Generalized Circumradius

TL;DR

This paper explores the generalized circumradius concept relative to convex bodies, characterizes the functions that can be represented this way, and connects it with the theory of diversities, providing specific characterizations for certain convex bodies.

Contribution

It introduces a characterization of functions arising from the generalized circumradius with respect to convex bodies, linking it to diversities and providing explicit results for simplices and parallelotopes.

Findings

Characterization of functions from generalized circumradius.

Connection between circumradius functions and diversities.

Explicit characterizations for simplices and parallelotopes.

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 $\lambda \geq 0$ 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.