We study the asymptotic behavior of the maximum interpoint distance of random
points in a d-dimensional set with a unique diameter and a smooth boundary at
the poles. Instead of investigating only a fixed number of n points as n
tends to infinity, we consider the much more general setting in which the
random points are the supports of appropriately defined Poisson processes. The
main result covers the case of uniformly distributed points within a
d-dimensional ellipsoid with a unique major axis. Moreover, several
generalizations of the main result are established, for example a limit law for
the maximum interpoint distance of random points from a Pearson type II
distribution
