Abstract
The
generalized circumradius
of a set of points
A
⊆
R
d
with respect to a convex body
K
equals the minimum value of
λ
≥
0
such that a translate of
λ
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.