  We study the asymptotic behavior of the diameter or maximum interpoint
distance of a cloud of i.i.d. $d$-dimensional random vectors when the number of
points in the cloud tends to infinity. This is a non standard extreme value
problem since the diameter is a max-$U$-statistic, hence a maximum of dependent
random variables. Therefore, the limiting distributions may not be extreme
value distributions. We obtain exhaustive results for the Euclidean diameter of
a cloud of elliptical vectors whose Euclidean norm is in the domain of
attraction for the maximum of the Gumbel distribution. We also obtain results
in other norms for spherical vectors and we give several bi-dimensional
generalizations. The main idea behind our results and their proofs is a
specific property of random vectors whose norm is in the domain of attraction
of the Gumbel distribution: the localization into subspaces of low dimension of
vectors with a large norm.
