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Full characterization of optimal transport plans for concave costs
1. | Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, 91405 Orsay cedex, France |
2. | Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay cedex |
3. | Cambridge Centre for Analysis, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom |
References:
[1] |
G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in $\mathbbR^n$, Math. Z., 230 (1999), 259-316.
doi: 10.1007/PL00004691. |
[2] |
Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, 44 (1991), 375-417.
doi: 10.1002/cpa.3160440402. |
[3] |
T. Champion and L. De Pascale, The Monge problem in $R^d$, Duke Math. J., 157 (2011), 551-572.
doi: 10.1215/00127094-1272939. |
[4] |
T. Champion and L. De Pascale, On the twist condition and $c$-monotone transport plans, Discr. Cont. Dyn. Syst. Ser. A, 34 (2014), 1339-1353. |
[5] |
T. Champion, L. De Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. of Mathematical Analysis, 40 (2008), 1-20.
doi: 10.1137/07069938X. |
[6] |
J. Delon, J. Salomon and A. Sobolevskii, Local matching indicators for transport problems with concave costs, SIAM J. Disc. Math., 26 (2012), 801-827.
doi: 10.1137/110823304. |
[7] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. |
[8] |
H. Federer, Geometric Measure Theory, Classics in Mathematics, Springer, 1996.
doi: 10.1007/978-3-642-62010-2. |
[9] |
W. Gangbo and R. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161.
doi: 10.1007/BF02392620. |
[10] |
L. V. Kantorovich, On the translocation of masses, C. R. (Dokl.) Acad. Sci. URSS, 37 (1942), 199-201. |
[11] |
L. V. Kantorovich, On a problem of Monge (Russian), Uspekhi Mat. Nauk., 3 (1948), 225-226. |
[12] |
X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.
doi: 10.1007/s00205-005-0362-9. |
[13] |
G. Monge, Mémoire sur la théorie des Déblais et des Remblais (French), Histoire de l'Académie des Sciences de Paris, 1781. |
[14] |
A. Pratelli, On the sufficiency of c-cyclical monotonicity for optimality of transport plans, Math. Z., 258 (2008), 677-690.
doi: 10.1007/s00209-007-0191-7. |
[15] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, AMS, 2003. |
show all references
References:
[1] |
G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in $\mathbbR^n$, Math. Z., 230 (1999), 259-316.
doi: 10.1007/PL00004691. |
[2] |
Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, 44 (1991), 375-417.
doi: 10.1002/cpa.3160440402. |
[3] |
T. Champion and L. De Pascale, The Monge problem in $R^d$, Duke Math. J., 157 (2011), 551-572.
doi: 10.1215/00127094-1272939. |
[4] |
T. Champion and L. De Pascale, On the twist condition and $c$-monotone transport plans, Discr. Cont. Dyn. Syst. Ser. A, 34 (2014), 1339-1353. |
[5] |
T. Champion, L. De Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. of Mathematical Analysis, 40 (2008), 1-20.
doi: 10.1137/07069938X. |
[6] |
J. Delon, J. Salomon and A. Sobolevskii, Local matching indicators for transport problems with concave costs, SIAM J. Disc. Math., 26 (2012), 801-827.
doi: 10.1137/110823304. |
[7] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. |
[8] |
H. Federer, Geometric Measure Theory, Classics in Mathematics, Springer, 1996.
doi: 10.1007/978-3-642-62010-2. |
[9] |
W. Gangbo and R. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161.
doi: 10.1007/BF02392620. |
[10] |
L. V. Kantorovich, On the translocation of masses, C. R. (Dokl.) Acad. Sci. URSS, 37 (1942), 199-201. |
[11] |
L. V. Kantorovich, On a problem of Monge (Russian), Uspekhi Mat. Nauk., 3 (1948), 225-226. |
[12] |
X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.
doi: 10.1007/s00205-005-0362-9. |
[13] |
G. Monge, Mémoire sur la théorie des Déblais et des Remblais (French), Histoire de l'Académie des Sciences de Paris, 1781. |
[14] |
A. Pratelli, On the sufficiency of c-cyclical monotonicity for optimality of transport plans, Math. Z., 258 (2008), 677-690.
doi: 10.1007/s00209-007-0191-7. |
[15] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, AMS, 2003. |
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