Isodiametric sets in the Heisenberg group

TL;DR

This paper investigates the shape and properties of isodiametric sets in the Heisenberg group, revealing boundary regularity, characterizations under symmetry, and surprising non-uniqueness phenomena.

Contribution

It provides a detailed geometric analysis of isodiametric sets in the Heisenberg group, including boundary descriptions and characterizations of rotationally invariant cases.

Findings

Boundary of isodiametric sets is given by graphs of Lipschitz functions.

Rotationally invariant isodiametric sets' symmetrization matches the convex hull of a CC-ball.

Non-uniqueness results for isodiametric sets are established.

Abstract

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.