Remarks on multi-marginal symmetric Monge-Kantorovich problems

Nassif Ghoussoub; Bernard Maurey

doi:10.3934/dcds.2014.34.1465

Article Contents

2014, Volume 34, Issue 4: 1465-1480. Doi: 10.3934/dcds.2014.34.1465

This issue Previous Article Metric cycles, curves and solenoids Next Article The Abresch-Gromoll inequality in a non-smooth setting

Remarks on multi-marginal symmetric Monge-Kantorovich problems

Nassif Ghoussoub^1, and
Bernard Maurey^2,

1.
Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2
2.
Institut de Mathématiques, UMR 7586 - CNRS, Université Paris Diderot - Paris 7, Paris

Received: October 31, 2012

Revised: January 31, 2013

Published: March 2014

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Abstract

Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Święch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.

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Mathematics Subject Classification: Primary: 49J30, 49K30, 49J52; Secondary: 58C35, 58E17, 91B68.

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