  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.
