Isodiametric sets in the Heisenberg group

  • Gian Paolo Leonardi

    Università di Modena e Reggio Emilia, Italy
  • Séverine Rigot

    Université de Nice Sophia Antipolis, France
  • Davide Vittone

    Università di Padova, Italy


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 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, within the restricted class of rotationally invariant sets, we give a quite complete characterization of any compact (rotationally invariant) isodiametric set. More precisely, its Steiner symmetrization with respect to the -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.

Cite this article

Gian Paolo Leonardi, Séverine Rigot, Davide Vittone, Isodiametric sets in the Heisenberg group. Rev. Mat. Iberoam. 28 (2012), no. 4, pp. 999–1024

DOI 10.4171/RMI/700