Revista Matemática Iberoamericana

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Volume 28, Issue 4, 2012, pp. 999–1024
DOI: 10.4171/RMI/700

Published online: 2012-10-14

Isodiametric sets in the Heisenberg group

Gian Paolo Leonardi[1], Séverine Rigot[2] and Davide Vittone[3]

(1) Università di Modena e Reggio Emilia, Italy
(2) Université de Nice Sophia Antipolis, France
(3) Università di Padova, Italy

In the sub-Riemannian Heisenberg group equipped with its Carnot--Carath\'eodory metric and with a Haar measure, we consider \textit{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 $\mathbb{C}^n$-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.

Keywords: Isodiametric problem, Heisenberg group

Leonardi Gian Paolo, Rigot Séverine, Vittone Davide: Isodiametric sets in the Heisenberg group. Rev. Mat. Iberoam. 28 (2012), 999-1024. doi: 10.4171/RMI/700