Isodiametric sets in the Heisenberg group

In the sub-Riemannian Heisenberg group equipped with its Carnot-Carathéodory 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 C^n-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.