Methods and Applications of Analysis

Volume 28 (2021)

Number 1

Special Issue for the 60th Birthday of John Urbas: Part II

Guest Editors: Neil Trudinger and Xu-Jia Wang

Multi-marginal optimal transport on the Heisenberg group

Pages: 67 – 76

DOI:  https://dx.doi.org/10.4310/MAA.2021.v28.n1.a5

Authors

Brendan Pass (Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada)

Andrea Pinamonti (Dipartimento di Matematica, Università degli Studi di Trento, Italy)

Mattia Vedovato (Dipartimento di Matematica, Università degli Studi di Trento, Italy)

Abstract

We consider the multi-marginal optimal transport of aligning several compactly supported marginals on the Heisenberg group to minimize the total cost, which we take to be the sum of the squared Carnot–Carathéodory distances from the marginal points to their barycenter. Under certain technical hypotheses, we prove existence and uniqueness of optimal maps. We also point out several related open questions.

Keywords

optimal transport, multi-marginal problems, Heisenberg group, sub-Riemannian geometry, Wasserstein barycenters

2010 Mathematics Subject Classification

35R03, 49Qxx, 53C17

B.P. is pleased to acknowledge support from Natural Sciences and Engineering Research Council of Canada Grant 04658-2018.

A.P. is partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INDAM).

Received 21 March 2020

Accepted 11 June 2020

Published 1 December 2021