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December  2015, 35(12): 6113-6132. doi: 10.3934/dcds.2015.35.6113

Full characterization of optimal transport plans for concave costs

Paul Pegon 1, , Filippo Santambrogio 2, and  Davide Piazzoli 3,

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

Received  November 2013 Published  May 2015

This paper slightly improves a classical result by Gangbo and McCann (1996) about the structure of optimal transport plans for costs that are strictly concave and increasing functions of the Euclidean distance. Since the main difficulty for proving the existence of an optimal map comes from the possible singularity of the cost at $0$, everything is quite easy if the supports of the two measures are disjoint; Gangbo and McCann proved the result under the assumption $\mu(supp(\mathbf{v}))=0$; in this paper we replace this assumption with the fact that the two measures are singular to each other. In this case it is possible to prove the existence of an optimal transport map, provided the starting measure $\mu$ does not give mass to small sets (i.e. $(d\!-\!1)-$rectifiable sets). When the measures are not singular the optimal transport plan decomposes into two parts, one concentrated on the diagonal and the other being a transport map between mutually singular measures.
Keywords: Monge-Kantorovich, rectifiable sets, approximate gradient, density points., transport maps.
Mathematics Subject Classification: 49J45, 49K21, 49Q20, 28A7.
Citation: Paul Pegon, Filippo Santambrogio, Davide Piazzoli. Full characterization of optimal transport plans for concave costs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6113-6132. doi: 10.3934/dcds.2015.35.6113
References:
[1]

G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in $\mathbbR^n$,, Math. Z., 230 (1999), 259. doi: 10.1007/PL00004691. Google Scholar

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Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions,, Communications on Pure and Applied Mathematics, 44 (1991), 375. doi: 10.1002/cpa.3160440402. Google Scholar

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T. Champion and L. De Pascale, The Monge problem in $R^d$,, Duke Math. J., 157 (2011), 551. doi: 10.1215/00127094-1272939. Google Scholar

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T. Champion and L. De Pascale, On the twist condition and $c$-monotone transport plans,, Discr. Cont. Dyn. Syst. Ser. A, 34 (2014), 1339. Google Scholar

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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. doi: 10.1137/07069938X. Google Scholar

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J. Delon, J. Salomon and A. Sobolevskii, Local matching indicators for transport problems with concave costs,, SIAM J. Disc. Math., 26 (2012), 801. doi: 10.1137/110823304. Google Scholar

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L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar

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H. Federer, Geometric Measure Theory,, Classics in Mathematics, (1996). doi: 10.1007/978-3-642-62010-2. Google Scholar

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W. Gangbo and R. McCann, The geometry of optimal transportation,, Acta Math., 177 (1996), 113. doi: 10.1007/BF02392620. Google Scholar

[10]

L. V. Kantorovich, On the translocation of masses,, C. R. (Dokl.) Acad. Sci. URSS, 37 (1942), 199. Google Scholar

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L. V. Kantorovich, On a problem of Monge (Russian),, Uspekhi Mat. Nauk., 3 (1948), 225. Google Scholar

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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. doi: 10.1007/s00205-005-0362-9. Google Scholar

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G. Monge, Mémoire sur la théorie des Déblais et des Remblais (French),, Histoire de l'Académie des Sciences de Paris, (1781). Google Scholar

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A. Pratelli, On the sufficiency of c-cyclical monotonicity for optimality of transport plans,, Math. Z., 258 (2008), 677. doi: 10.1007/s00209-007-0191-7. Google Scholar

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show all references

References:
[1]

G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in $\mathbbR^n$,, Math. Z., 230 (1999), 259. doi: 10.1007/PL00004691. Google Scholar

[2]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions,, Communications on Pure and Applied Mathematics, 44 (1991), 375. doi: 10.1002/cpa.3160440402. Google Scholar

[3]

T. Champion and L. De Pascale, The Monge problem in $R^d$,, Duke Math. J., 157 (2011), 551. doi: 10.1215/00127094-1272939. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/07069938X. Google Scholar

[6]

J. Delon, J. Salomon and A. Sobolevskii, Local matching indicators for transport problems with concave costs,, SIAM J. Disc. Math., 26 (2012), 801. doi: 10.1137/110823304. Google Scholar

[7]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar

[8]

H. Federer, Geometric Measure Theory,, Classics in Mathematics, (1996). doi: 10.1007/978-3-642-62010-2. Google Scholar

[9]

W. Gangbo and R. McCann, The geometry of optimal transportation,, Acta Math., 177 (1996), 113. doi: 10.1007/BF02392620. Google Scholar

[10]

L. V. Kantorovich, On the translocation of masses,, C. R. (Dokl.) Acad. Sci. URSS, 37 (1942), 199. Google Scholar

[11]

L. V. Kantorovich, On a problem of Monge (Russian),, Uspekhi Mat. Nauk., 3 (1948), 225. Google Scholar

[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. doi: 10.1007/s00205-005-0362-9. Google Scholar

[13]

G. Monge, Mémoire sur la théorie des Déblais et des Remblais (French),, Histoire de l'Académie des Sciences de Paris, (1781). Google Scholar

[14]

A. Pratelli, On the sufficiency of c-cyclical monotonicity for optimality of transport plans,, Math. Z., 258 (2008), 677. doi: 10.1007/s00209-007-0191-7. Google Scholar

[15]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003). Google Scholar

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