Severine Rigot

Optimal mass transportation in the Heisenberg group

In this paper we consider the problem of optimal transportation of absolutely continuous masses in the Heisenberg group Hn, in the case when the cost function is either the square of the Carnot–Caratheodory distance or the square of the Koranyi norm. In both cases we show existence and uniqueness of an optimal transport map. In the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function. In the latter case the proof requires some fine properties of BV functions in the Heisenberg group.

MASS TRANSPORTATION IN GROUPS OF TYPE H

We give a solution of the optimal transport problem in groups of type H when the cost function is the square of either the Carnot–Caratheodory or Koranyi distance. This generalizes results previously proved for the Heisenberg groups. We use the same strategy that the one which was developed in that special case together with slightly refined technicalities that essentially reflect the fact that the center of the group can be of dimension larger than one. For each distance we prove existence, uniqueness and give a characterization of the optimal transport. In the case of the Carnot–Caratheodory distance we also prove that the optimal transport arises as the limit of the optimal transports in natural Riemannian approximations.

Besicovitch covering property for homogeneous distances on the Heisenberg groups

Our main result is a positive answer to the question whether one can find homogeneous distances on the Heisenberg groups that have the Besicovitch Covering Property (BCP). This property is well known to be one of the fundamental tools of measure theory, with strong connections with the theory of differentiation of measures. We prove that BCP is satisfied by the homogeneous distances whose unit ball centered at the origin coincides with an Euclidean ball. Such homogeneous distances do exist on any Carnot group by a result of Hebisch and Sikora. In the Heisenberg groups, they are related to the Cygan-Koranyi (also called Koranyi) distance. They were considered in particular by Lee and Naor to provide a counterexample to the Goemans-Linial conjecture in theoretical computer science. To put our result in perspective, we also prove two geometric criteria that imply the non-validity of BCP, showing that in some sense our example is sharp. Our first criterion applies in particular to commonly used homogeneous distances on the Heisenberg groups, such as the Cygan-Koranyi and Carnot-Caratheodory distances that are already known not to satisfy BCP. To put a different perspective on these results and for sake of completeness, we also give a proof of the fact, noticed by D. Preiss, that in a general metric space, one can always construct a bi-Lipschitz equivalent distance that does not satisfy BCP.

Monge's transport problem in the Heisenberg group

Abstract We prove the existence of a solution to Monges transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot–Carathéodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group.

Isodiametric sets in the Heisenberg group

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.

Quasiminimal Partitions with Prescribed Measure

We prove that, in a fairly general context, quasiminimal partitions with prescribed measure enjoy quantitative rectifiability properties with universal bounds. Namely, we show that the set of interfaces of a quasiminimal partition is uniformly rectifiable with bounds that depend only on the structural data of the problem.