Full characterization of optimal transport plans for concave costs

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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

Abstract
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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 and Continuous Dynamical Systems, 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 $\mathbb{R}^{N}$, 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 $mathbb{R}^{d}$, 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 $\mathbb{R}^{N}$, 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 $mathbb{R}^{d}$, 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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