April  2014, 34(4): 1605-1621. doi: 10.3934/dcds.2014.34.1605

A glimpse into the differential topology and geometry of optimal transport

1. 

Department of Mathematics, University of Toronto, Toronto ON Canada M5R 2Y4, Canada

Received  July 2012 Revised  November 2012 Published  October 2013

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.
Citation: Robert J. McCann. A glimpse into the differential topology and geometry of optimal transport. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605
References:
[1]

N. Ahmad, "The Geometry of Shape Recognition Via a Monge-Kantorovich Optimal Transport Problem,'' PhD thesis, Brown University, 2004. Available from http://www.math.toronto.edu/mccann/ahmad.pdf

[2]

N. Ahmad, H. K. Kim and R.J. McCann, Optimal transportation, topology and uniqueness, Bull. Math. Sci., 1 (2011), 13-32. doi: 10.1007/s13373-011-0002-7.

[3]

G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in $\mathbbR^n$, Math. Z., 230 (1999), 259-316. doi: 10.1007/PL00004691.

[4]

L. Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces (Funchal, 2000),'' Springer Lecture Notes in Mathematics, 1812 (2003), 1-52. doi: 10.1007/978-3-540-39189-0_1.

[5]

L. A. Ambrosio and N. Gigli, A user's guide to optimal transport,, Preprint., ().  doi: 10.1007/978-3-642-32160-3_1.

[6]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.

[7]

J.-P. Bourguignon, Ricci curvature and measures, Japan. J. Math., 4 (2009), 27-45. doi: 10.1007/s11537-009-0855-7.

[8]

Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, (French) [Polar decomposition and monotone rearrangement of vector fields] C.R. Acad. Sci. Paris Sér. I Math., 305 (1987), 805-808.

[9]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.

[10]

L. A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104. doi: 10.1090/S0894-0347-1992-1124980-8.

[11]

L. A. Caffarelli, Boundary regularity of maps with convex potentials - II, Ann. of Math. (2), 144 (1996), 453-496. doi: 10.2307/2118564.

[12]

L. A. Caffarelli, M. Feldman and R. J. McCann, Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs, J. Amer. Math. Soc., 15 (2002), 1-26. doi: 10.1090/S0894-0347-01-00376-9.

[13]

T. Champion and L. De Pascale, The Monge problem in $\mathbbR^d$, Duke Math. J., 157 (2011), 551-572. doi: 10.1215/00127094-1272939.

[14]

P.-A. Chiappori, R. J. McCann and L. Nesheim, Hedonic price equilibria, stable matching and optimal transport: Equivalence, topology and uniqueness, Econom. Theory, 42 (2010), 317-354. doi: 10.1007/s00199-009-0455-z.

[15]

D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257. doi: 10.1007/s002220100160.

[16]

M. J. P Cullen and R. J. Purser, An extended Lagrangian model of semi-geostrophic frontogenesis, J. Atmos. Sci., 41 (1984), 1477-1497. doi: 10.1175/1520-0469(1984)041<1477:AELTOS>2.0.CO;2.

[17]

P. Delanoë, Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator, Ann. Inst. H. Poincarè Anal. Non Linèaire, 8 (1991), 443-457. doi: 10.1016/j.anihpc.2007.03.001.

[18]

P. Delanoë and Y. Ge, Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds, J. Reine Angew. Math., 646 (2010), 65-115. doi: 10.1515/CRELLE.2010.066.

[19]

P. Delanoë and F. Rouvière, Positively curved Riemannian locally symmetric spaces are positively square distance curved, Canad. J. Math. 65 (2013), 757-767. doi: 10.4153/CJM-2012-015-1.

[20]

R. M. Dudley, "Probabilities and Metrics - Convergence of Laws on Metric Spaces, with A View to Statistical Testing,'' Lecture Notes Series, No. 45. Matematisk Institut, Aarhus Universitet, Aarhus, 1976. ii+126 pp. (not consecutively paged).

[21]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), 1-66. doi: 10.1090/memo/0653.

[22]

M. Feldman and R.J. McCann, Uniqueness and transport density in Monge's transportation problem, Calc. Var. Partial Differential Equations, 15 (2002), 81-113. doi: 10.1007/s005260100119.

[23]

A. Figalli, Regularity properties of optimal maps between nonconvex domains in the plane, Comm. Partial Differential Equations, 35 (2010), 465-479. doi: 10.1080/03605300903307673.

[24]

A. Figalli and Y.-H. Kim, Partial regularity of Brenier solutions of the Monge-Ampère equation, Discrete Contin. Dyn. Syst., 28 (2010), 559-565. doi: 10.3934/dcds.2010.28.559.

[25]

A. Figalli, Y.-H. Kim and R. J. McCann, Hölder continuity and injectivity of optimal maps, Arch. Rational Mech. Anal., 209 (2013), 747-795. doi: 10.1007/s00205-013-0629-5.

[26]

A. Figalli, Y.-H. Kim and R. J. McCann, Regularity of optimal transport maps on multiple products of spheres, J. Euro. Math. Soc., (JEMS) 15 (2013), 1131-1166. doi: 10.4171/JEMS/388.

[27]

A. Figalli, Y.-H. Kim and R. J. McCann, When is multidimensional screening a convex program?, J. Econom Theory, 146 (2011), 454-478. doi: 10.1016/j.jet.2010.11.006.

[28]

A. Figalli and L. Rifford, Continuity of optimal transport maps on small deformations of $\mathbbS^2$, Comm. Pure Appl. Math., 62 (2009), 1670-1706. doi: 10.1002/cpa.20293.

[29]

A. Figalli, L. Rifford and C. Villani, Nearly round spheres look convex, Amer. J. Math., 134 (2012), 109-139. doi: 10.1353/ajm.2012.0000.

[30]

L. Forzani and D. Maldonado, Properties of the solutions to the Monge-Ampère equation, Nonlinear Anal., 57 (2004), 815-829. doi: 10.1016/j.na.2004.03.019.

[31]

W. Gangbo., "Habilitation Thesis," Université de Metz, 1995.

[32]

W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620.

[33]

W. Gangbo and R. J. McCann, Shape recognition via Wasserstein distance, Quart. Appl. Math., 58 (2000), 705-737.

[34]

N. Gigli, On the inverse implication of Brenier-McCann theorems and the structure of $(P_2(M),W_2)$, Methods Appl. Anal., 18 (2011), 127-158.

[35]

F. R. Harvey and H. B. Lawson, Jr, Split special Lagrangian geometry, Progress in Mathematics 297 (2012), 43-89. doi: 10.1007/978-3-0348-0257-4_3.

[36]

K. Hestir and S. C. Williams, Supports of doubly stochastic measures, Bernoulli, 1 (1995), 217-243. doi: 10.2307/3318478.

[37]

L. Kantorovich, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N. S.), 37 (1942), 199-201.

[38]

Y.-H. Kim and R. J. McCann, Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular), J. Reine Angew. Math., 664 (2012), 1-27. doi: 10.1515/CRELLE.2011.105.

[39]

Y.-H. Kim and R. J. McCann, Continuity, curvature, and the general covariance of optimal transportation, J. Eur. Math. Soc. (JEMS), 12 (2010), 1009-1040. doi: 10.4171/JEMS/221.

[40]

Y.-H. Kim, R. J. McCann and M. Warren, Pseudo-Riemannian geometry calibrates optimal transportation, Math. Res. Lett., 17 (2010), 1183-1197.

[41]

J. Kitagawa and M. Warren, Regularity for the optimal transportation problem with Euclidean distance squared cost on the embedded sphere, SIAM J. Math. Anal., 44 (2012), 2871-2887. doi: 10.1137/120865409.

[42]

T. C. Koopmans and M. Beckmann, Assignment problems and the location of economic activities, Econometrica, 25 (1957), 53-76. doi: 10.2307/1907742.

[43]

P. W. Y. Lee, New computable necessary conditions for the regularity theory of optimal transportation, SIAM J. Math. Anal., 42 (2010), 3054-3075. doi: 10.1137/100797722.

[44]

P. W. Y. Lee and J. Li, New examples on spaces of negative sectional curvature satisfying Ma-Trudinger-Wang conditions, SIAM J. Math. Anal., 44 (2012), 61-73. doi: 10.1137/110820543.

[45]

P. W. Y. Lee and R. J. McCann, The Ma-Trudinger-Wang curvature for natural mechanical actions, Calc. Var. Partial Differential Equations, 41 (2011), 285-299. doi: 10.1007/s00526-010-0362-y.

[46]

V. L. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem, Set-valued Anal., 7 (1999), 7-32. doi: 10.1023/A:1008753021652.

[47]

J. Li, "Smooth Optimal Transportation on Hyperbolic Space,'' Master's thesis, University of Toronto, 2009. Available at http://www.math.toronto.edu/mccann/papers/Li.pdf.

[48]

J. Liu, Hölder regularity of optimal mappings in optimal transportation, Calc Var. Partial Differential Equations, 34 (2009), 435-451. doi: 10.1007/s00526-008-0190-5.

[49]

J. Liu, N. S. Trudinger X.-J. Wang, Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184. doi: 10.1080/03605300903236609.

[50]

G. Loeper, On the regularity of solutions of optimal transportation problems, Acta Math., 202 (2009), 241-283. doi: 10.1007/s11511-009-0037-8.

[51]

G. Loeper, Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna, Arch. Ration. Mech. Anal., 199 (2011), 269-289. doi: 10.1007/s00205-010-0330-x.

[52]

G. Loeper and C. Villani, Regularity of optimal transport in curved geometry: The non-focal case, Duke Math. J., 151 (2010), 431-485. doi: 10.1215/00127094-2010-003.

[53]

G. G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly, 60 (1953), 176-179. doi: 10.2307/2307574.

[54]

J. Lott and C. Villani, Ricci curvature for metric measure spaces via optimal transport, Annals Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903.

[55]

X.-N. Ma, N. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Rational Mech. Anal., 177 (2005), 151-183. doi: 10.1007/s00205-005-0362-9.

[56]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323. doi: 10.1215/S0012-7094-95-08013-2.

[57]

R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.

[58]

R. J. McCann, Exact solutions to the transportation problem on the line, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 1341-1380. doi: 10.1098/rspa.1999.0364.

[59]

R. J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679.

[60]

R. J. McCann and N. Guillen, Five lectures on optimal transportation: geometry, regularity, and applications, in "Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the the Séminaire de Mathématiques Supérieure (SMS) held in Montréal, QC, June 27-July 8, 2011,'' (eds. G. Dafni et al), American Mathematical Society, (2013), 145-180. Available at arXiv:1011.2911

[61]

R. J. McCann and M. Sosio, Hölder continuity of optimal multivalued mappings, SIAM J. Math. Anal., 43 (2011), 1855-1871. doi: 10.1137/100802670.

[62]

R. J. McCann, B. Pass and M. Warren, Rectifiability of optimal transportation plans, Canad. J. Math, 64 (2012), 924-934. doi: 10.4153/CJM-2011-080-6.

[63]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2.

[64]

J. A. Mirrlees, An exploration in the theory of optimum income taxation, Rev. Econom. Stud., 38 (1971), 175-208.

[65]

G. Monge, Mémoire sur la théorie des déblais et de remblais, in "Histoire de l'Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la Même Année," (1781), 666-704.

[66]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.

[67]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.

[68]

B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem, Calc. Var. Partial Differential Equations, 43 (2012), 529-536. doi: 10.1007/s00526-011-0421-z.

[69]

A. Pratelli, On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation, Ann. Inst. H. Poincaré Probab. Statist., 43 (2007), 1-13. doi: 10.1016/j.anihpb.2005.12.001.

[70]

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.

[71]

S. T. Rachev and L. Rüschendorf., "Mass Transportation Problems," Vol. I. Theory. Probability and its Applications (New York). Springer-Verlag, New York, 1998.

[72]

M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940. doi: 10.1002/cpa.20060.

[73]

J.-C. Rochet, A necessary and sufficient condition for rationalizability in a quasi-linear context, J. Math. Econom., 16 (1987), 191-200. doi: 10.1016/0304-4068(87)90007-3.

[74]

R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510. doi: 10.2140/pjm.1966.17.497.

[75]

L. Rüschendorf and S.T. Rachev, A characterization of random variables with minimum $L^2$-distance, J. Multivariate Anal., 32 (1990), 48-54. doi: 10.1016/0047-259X(90)90070-X.

[76]

W. Schachermayer and J. Teichmann, Characterization of optimal transport plans for the Monge-Kantorovich problem, Proc. Amer. Math. Soc., 137 (2009), 519-529. doi: 10.1090/S0002-9939-08-09419-7.

[77]

C. Smith and M. Knott, Note on the optimal transportation of distributions, J. Optim. Theory Appl., 52 (1987), 323-329. doi: 10.1007/BF00941290.

[78]

M. Spence, Job market signaling, Quarterly J. Econom., 87 (1973), 355-374. doi: 10.2307/1882010.

[79]

K.-T. Sturm, On the geometry of metric measure spaces, I. , Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8.

[80]

N. S. Trudinger and X.-J. Wang, On the Monge mass transfer problem, Calc. Var. Paritial Differential Equations, 13 (2001), 19-31. doi: 10.1007/PL00009922.

[81]

N. S. Trudinger and X.-J. Wang, On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8 (2009), 143-174..

[82]

J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487 (1997), 115-124. doi: 10.1515/crll.1997.487.115.

[83]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[84]

C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. doi: 10.1007/978-3-540-71050-9.

[85]

X.-J. Wang, On the design of a reflector antenna, Inverse Problems, 12 (1996), 351-375. doi: 10.1088/0266-5611/12/3/013.

[86]

X.-J. Wang, On the design of a reflector antenna II, Calc. Var. Partial Differential Equations, 20 (2004), 329-341. doi: 10.1007/s00526-003-0239-4.

[87]

Y. Yu, Singular set of a convex potential in two dimensions, Comm. Partial Differential Equations, 32 (2007), 1883-1894. doi: 10.1080/03605300701318757.

show all references

References:
[1]

N. Ahmad, "The Geometry of Shape Recognition Via a Monge-Kantorovich Optimal Transport Problem,'' PhD thesis, Brown University, 2004. Available from http://www.math.toronto.edu/mccann/ahmad.pdf

[2]

N. Ahmad, H. K. Kim and R.J. McCann, Optimal transportation, topology and uniqueness, Bull. Math. Sci., 1 (2011), 13-32. doi: 10.1007/s13373-011-0002-7.

[3]

G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in $\mathbbR^n$, Math. Z., 230 (1999), 259-316. doi: 10.1007/PL00004691.

[4]

L. Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces (Funchal, 2000),'' Springer Lecture Notes in Mathematics, 1812 (2003), 1-52. doi: 10.1007/978-3-540-39189-0_1.

[5]

L. A. Ambrosio and N. Gigli, A user's guide to optimal transport,, Preprint., ().  doi: 10.1007/978-3-642-32160-3_1.

[6]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.

[7]

J.-P. Bourguignon, Ricci curvature and measures, Japan. J. Math., 4 (2009), 27-45. doi: 10.1007/s11537-009-0855-7.

[8]

Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, (French) [Polar decomposition and monotone rearrangement of vector fields] C.R. Acad. Sci. Paris Sér. I Math., 305 (1987), 805-808.

[9]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.

[10]

L. A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104. doi: 10.1090/S0894-0347-1992-1124980-8.

[11]

L. A. Caffarelli, Boundary regularity of maps with convex potentials - II, Ann. of Math. (2), 144 (1996), 453-496. doi: 10.2307/2118564.

[12]

L. A. Caffarelli, M. Feldman and R. J. McCann, Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs, J. Amer. Math. Soc., 15 (2002), 1-26. doi: 10.1090/S0894-0347-01-00376-9.

[13]

T. Champion and L. De Pascale, The Monge problem in $\mathbbR^d$, Duke Math. J., 157 (2011), 551-572. doi: 10.1215/00127094-1272939.

[14]

P.-A. Chiappori, R. J. McCann and L. Nesheim, Hedonic price equilibria, stable matching and optimal transport: Equivalence, topology and uniqueness, Econom. Theory, 42 (2010), 317-354. doi: 10.1007/s00199-009-0455-z.

[15]

D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257. doi: 10.1007/s002220100160.

[16]

M. J. P Cullen and R. J. Purser, An extended Lagrangian model of semi-geostrophic frontogenesis, J. Atmos. Sci., 41 (1984), 1477-1497. doi: 10.1175/1520-0469(1984)041<1477:AELTOS>2.0.CO;2.

[17]

P. Delanoë, Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator, Ann. Inst. H. Poincarè Anal. Non Linèaire, 8 (1991), 443-457. doi: 10.1016/j.anihpc.2007.03.001.

[18]

P. Delanoë and Y. Ge, Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds, J. Reine Angew. Math., 646 (2010), 65-115. doi: 10.1515/CRELLE.2010.066.

[19]

P. Delanoë and F. Rouvière, Positively curved Riemannian locally symmetric spaces are positively square distance curved, Canad. J. Math. 65 (2013), 757-767. doi: 10.4153/CJM-2012-015-1.

[20]

R. M. Dudley, "Probabilities and Metrics - Convergence of Laws on Metric Spaces, with A View to Statistical Testing,'' Lecture Notes Series, No. 45. Matematisk Institut, Aarhus Universitet, Aarhus, 1976. ii+126 pp. (not consecutively paged).

[21]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), 1-66. doi: 10.1090/memo/0653.

[22]

M. Feldman and R.J. McCann, Uniqueness and transport density in Monge's transportation problem, Calc. Var. Partial Differential Equations, 15 (2002), 81-113. doi: 10.1007/s005260100119.

[23]

A. Figalli, Regularity properties of optimal maps between nonconvex domains in the plane, Comm. Partial Differential Equations, 35 (2010), 465-479. doi: 10.1080/03605300903307673.

[24]

A. Figalli and Y.-H. Kim, Partial regularity of Brenier solutions of the Monge-Ampère equation, Discrete Contin. Dyn. Syst., 28 (2010), 559-565. doi: 10.3934/dcds.2010.28.559.

[25]

A. Figalli, Y.-H. Kim and R. J. McCann, Hölder continuity and injectivity of optimal maps, Arch. Rational Mech. Anal., 209 (2013), 747-795. doi: 10.1007/s00205-013-0629-5.

[26]

A. Figalli, Y.-H. Kim and R. J. McCann, Regularity of optimal transport maps on multiple products of spheres, J. Euro. Math. Soc., (JEMS) 15 (2013), 1131-1166. doi: 10.4171/JEMS/388.

[27]

A. Figalli, Y.-H. Kim and R. J. McCann, When is multidimensional screening a convex program?, J. Econom Theory, 146 (2011), 454-478. doi: 10.1016/j.jet.2010.11.006.

[28]

A. Figalli and L. Rifford, Continuity of optimal transport maps on small deformations of $\mathbbS^2$, Comm. Pure Appl. Math., 62 (2009), 1670-1706. doi: 10.1002/cpa.20293.

[29]

A. Figalli, L. Rifford and C. Villani, Nearly round spheres look convex, Amer. J. Math., 134 (2012), 109-139. doi: 10.1353/ajm.2012.0000.

[30]

L. Forzani and D. Maldonado, Properties of the solutions to the Monge-Ampère equation, Nonlinear Anal., 57 (2004), 815-829. doi: 10.1016/j.na.2004.03.019.

[31]

W. Gangbo., "Habilitation Thesis," Université de Metz, 1995.

[32]

W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620.

[33]

W. Gangbo and R. J. McCann, Shape recognition via Wasserstein distance, Quart. Appl. Math., 58 (2000), 705-737.

[34]

N. Gigli, On the inverse implication of Brenier-McCann theorems and the structure of $(P_2(M),W_2)$, Methods Appl. Anal., 18 (2011), 127-158.

[35]

F. R. Harvey and H. B. Lawson, Jr, Split special Lagrangian geometry, Progress in Mathematics 297 (2012), 43-89. doi: 10.1007/978-3-0348-0257-4_3.

[36]

K. Hestir and S. C. Williams, Supports of doubly stochastic measures, Bernoulli, 1 (1995), 217-243. doi: 10.2307/3318478.

[37]

L. Kantorovich, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N. S.), 37 (1942), 199-201.

[38]

Y.-H. Kim and R. J. McCann, Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular), J. Reine Angew. Math., 664 (2012), 1-27. doi: 10.1515/CRELLE.2011.105.

[39]

Y.-H. Kim and R. J. McCann, Continuity, curvature, and the general covariance of optimal transportation, J. Eur. Math. Soc. (JEMS), 12 (2010), 1009-1040. doi: 10.4171/JEMS/221.

[40]

Y.-H. Kim, R. J. McCann and M. Warren, Pseudo-Riemannian geometry calibrates optimal transportation, Math. Res. Lett., 17 (2010), 1183-1197.

[41]

J. Kitagawa and M. Warren, Regularity for the optimal transportation problem with Euclidean distance squared cost on the embedded sphere, SIAM J. Math. Anal., 44 (2012), 2871-2887. doi: 10.1137/120865409.

[42]

T. C. Koopmans and M. Beckmann, Assignment problems and the location of economic activities, Econometrica, 25 (1957), 53-76. doi: 10.2307/1907742.

[43]

P. W. Y. Lee, New computable necessary conditions for the regularity theory of optimal transportation, SIAM J. Math. Anal., 42 (2010), 3054-3075. doi: 10.1137/100797722.

[44]

P. W. Y. Lee and J. Li, New examples on spaces of negative sectional curvature satisfying Ma-Trudinger-Wang conditions, SIAM J. Math. Anal., 44 (2012), 61-73. doi: 10.1137/110820543.

[45]

P. W. Y. Lee and R. J. McCann, The Ma-Trudinger-Wang curvature for natural mechanical actions, Calc. Var. Partial Differential Equations, 41 (2011), 285-299. doi: 10.1007/s00526-010-0362-y.

[46]

V. L. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem, Set-valued Anal., 7 (1999), 7-32. doi: 10.1023/A:1008753021652.

[47]

J. Li, "Smooth Optimal Transportation on Hyperbolic Space,'' Master's thesis, University of Toronto, 2009. Available at http://www.math.toronto.edu/mccann/papers/Li.pdf.

[48]

J. Liu, Hölder regularity of optimal mappings in optimal transportation, Calc Var. Partial Differential Equations, 34 (2009), 435-451. doi: 10.1007/s00526-008-0190-5.

[49]

J. Liu, N. S. Trudinger X.-J. Wang, Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184. doi: 10.1080/03605300903236609.

[50]

G. Loeper, On the regularity of solutions of optimal transportation problems, Acta Math., 202 (2009), 241-283. doi: 10.1007/s11511-009-0037-8.

[51]

G. Loeper, Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna, Arch. Ration. Mech. Anal., 199 (2011), 269-289. doi: 10.1007/s00205-010-0330-x.

[52]

G. Loeper and C. Villani, Regularity of optimal transport in curved geometry: The non-focal case, Duke Math. J., 151 (2010), 431-485. doi: 10.1215/00127094-2010-003.

[53]

G. G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly, 60 (1953), 176-179. doi: 10.2307/2307574.

[54]

J. Lott and C. Villani, Ricci curvature for metric measure spaces via optimal transport, Annals Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903.

[55]

X.-N. Ma, N. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Rational Mech. Anal., 177 (2005), 151-183. doi: 10.1007/s00205-005-0362-9.

[56]

R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323. doi: 10.1215/S0012-7094-95-08013-2.

[57]

R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.

[58]

R. J. McCann, Exact solutions to the transportation problem on the line, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 1341-1380. doi: 10.1098/rspa.1999.0364.

[59]

R. J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679.

[60]

R. J. McCann and N. Guillen, Five lectures on optimal transportation: geometry, regularity, and applications, in "Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the the Séminaire de Mathématiques Supérieure (SMS) held in Montréal, QC, June 27-July 8, 2011,'' (eds. G. Dafni et al), American Mathematical Society, (2013), 145-180. Available at arXiv:1011.2911

[61]

R. J. McCann and M. Sosio, Hölder continuity of optimal multivalued mappings, SIAM J. Math. Anal., 43 (2011), 1855-1871. doi: 10.1137/100802670.

[62]

R. J. McCann, B. Pass and M. Warren, Rectifiability of optimal transportation plans, Canad. J. Math, 64 (2012), 924-934. doi: 10.4153/CJM-2011-080-6.

[63]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2.

[64]

J. A. Mirrlees, An exploration in the theory of optimum income taxation, Rev. Econom. Stud., 38 (1971), 175-208.

[65]

G. Monge, Mémoire sur la théorie des déblais et de remblais, in "Histoire de l'Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la Même Année," (1781), 666-704.

[66]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.

[67]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.

[68]

B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem, Calc. Var. Partial Differential Equations, 43 (2012), 529-536. doi: 10.1007/s00526-011-0421-z.

[69]

A. Pratelli, On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation, Ann. Inst. H. Poincaré Probab. Statist., 43 (2007), 1-13. doi: 10.1016/j.anihpb.2005.12.001.

[70]

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.

[71]

S. T. Rachev and L. Rüschendorf., "Mass Transportation Problems," Vol. I. Theory. Probability and its Applications (New York). Springer-Verlag, New York, 1998.

[72]

M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940. doi: 10.1002/cpa.20060.

[73]

J.-C. Rochet, A necessary and sufficient condition for rationalizability in a quasi-linear context, J. Math. Econom., 16 (1987), 191-200. doi: 10.1016/0304-4068(87)90007-3.

[74]

R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510. doi: 10.2140/pjm.1966.17.497.

[75]

L. Rüschendorf and S.T. Rachev, A characterization of random variables with minimum $L^2$-distance, J. Multivariate Anal., 32 (1990), 48-54. doi: 10.1016/0047-259X(90)90070-X.

[76]

W. Schachermayer and J. Teichmann, Characterization of optimal transport plans for the Monge-Kantorovich problem, Proc. Amer. Math. Soc., 137 (2009), 519-529. doi: 10.1090/S0002-9939-08-09419-7.

[77]

C. Smith and M. Knott, Note on the optimal transportation of distributions, J. Optim. Theory Appl., 52 (1987), 323-329. doi: 10.1007/BF00941290.

[78]

M. Spence, Job market signaling, Quarterly J. Econom., 87 (1973), 355-374. doi: 10.2307/1882010.

[79]

K.-T. Sturm, On the geometry of metric measure spaces, I. , Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8.

[80]

N. S. Trudinger and X.-J. Wang, On the Monge mass transfer problem, Calc. Var. Paritial Differential Equations, 13 (2001), 19-31. doi: 10.1007/PL00009922.

[81]

N. S. Trudinger and X.-J. Wang, On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8 (2009), 143-174..

[82]

J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487 (1997), 115-124. doi: 10.1515/crll.1997.487.115.

[83]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[84]

C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. doi: 10.1007/978-3-540-71050-9.

[85]

X.-J. Wang, On the design of a reflector antenna, Inverse Problems, 12 (1996), 351-375. doi: 10.1088/0266-5611/12/3/013.

[86]

X.-J. Wang, On the design of a reflector antenna II, Calc. Var. Partial Differential Equations, 20 (2004), 329-341. doi: 10.1007/s00526-003-0239-4.

[87]

Y. Yu, Singular set of a convex potential in two dimensions, Comm. Partial Differential Equations, 32 (2007), 1883-1894. doi: 10.1080/03605300701318757.

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