  In the sub-Riemannian Heisenberg group equipped with its Carnot-Caratheodory
metric and with a Haar measure, we consider isodiametric sets, i.e. sets
maximizing the measure among all sets with a given diameter. In particular,
given an isodiametric set, and up to negligible sets, we prove that its
boundary is given by the graphs of two locally Lipschitz functions. Moreover,
in the restricted class of rotationally invariant sets, we give a quite
complete characterization of any compact (rotationally invariant) isodiametric
set. More specifically, its Steiner symmetrization with respect to the Cn-plane
is shown to coincide with the Euclidean convex hull of a CC-ball. At the same
time, we also prove quite unexpected non-uniqueness results.
