# American Institute of Mathematical Sciences

April  2014, 34(4): 1465-1480. doi: 10.3934/dcds.2014.34.1465

## Remarks on multi-marginal symmetric Monge-Kantorovich problems

 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, France

Received  November 2012 Revised  February 2013 Published  October 2013

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.
Citation: Nassif Ghoussoub, Bernard Maurey. Remarks on multi-marginal symmetric Monge-Kantorovich problems. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1465-1480. doi: 10.3934/dcds.2014.34.1465
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##### References:
 [1] Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 853-865. doi: 10.3934/cpaa.2008.7.853 [2] Abbas Moameni. Invariance properties of the Monge-Kantorovich mass transport problem. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2653-2671. doi: 10.3934/dcds.2016.36.2653 [3] Giuseppe Buttazzo, Eugene Stepanov. Transport density in Monge-Kantorovich problems with Dirichlet conditions. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 607-628. doi: 10.3934/dcds.2005.12.607 [4] Zuo Quan Xu, Jia-An Yan. A note on the Monge-Kantorovich problem in the plane. Communications on Pure & Applied Analysis, 2015, 14 (2) : 517-525. doi: 10.3934/cpaa.2015.14.517 [5] Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $L_1$ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037 [6] Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial & Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431 [7] Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $(n, m)$-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117 [8] Julio C. Rebelo, Ana L. Silva. On the Burnside problem in Diff(M). Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 423-439. doi: 10.3934/dcds.2007.17.423 [9] David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311 [10] Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895 [11] Jun He, Guangjun Xu, Yanmin Liu. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. Journal of Industrial & Management Optimization, 2020, 16 (6) : 3035-3045. doi: 10.3934/jimo.2019092 [12] Zsolt Saffer, Wuyi Yue. M/M/c multiple synchronous vacation model with gated discipline. Journal of Industrial & Management Optimization, 2012, 8 (4) : 939-968. doi: 10.3934/jimo.2012.8.939 [13] Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1 [14] Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463 [15] Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008 [16] Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026 [17] Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481 [18] Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623 [19] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41 [20] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75

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