  Let a random geometric graph be defined in the supercritical regime for the
existence of a unique infinite connected component in Euclidean space. Consider
the first-passage percolation model with independent and identically
distributed random variables on the random infinite connected component. We
provide sufficient conditions for the existence of the asymptotic shape and we
show that the shape is an Euclidean ball. We give some examples exhibiting the
result for Bernoulli percolation and the Richardson model. For the Richardson
model we further show that it converges weakly to a nonstandard branching
process in the joint limit of large intensities and slow passage times.
