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 & Continuous Dynamical Systems - A, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605
References:
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N. Ahmad, "The Geometry of Shape Recognition Via a Monge-Kantorovich Optimal Transport Problem,'', PhD thesis, (2004).

[2]

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

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

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[9]

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

[10]

L. A. Caffarelli, The regularity of mappings with a convex potential,, J. Amer. Math. Soc., 5 (1992), 99. 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. 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. 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. 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. 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. doi: 10.1007/s002220100160.

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

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

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A. Figalli, Y.-H. Kim and R. J. McCann, Regularity of optimal transport maps on multiple products of spheres,, J. Euro. Math. Soc., 15 (2013), 1131. 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. doi: 10.1016/j.jet.2010.11.006.

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

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

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

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

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

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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.

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

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

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

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

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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. doi: 10.1137/120865409.

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

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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. doi: 10.1137/110820543.

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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. doi: 10.1007/s00526-010-0362-y.

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show all references

References:
[1]

N. Ahmad, "The Geometry of Shape Recognition Via a Monge-Kantorovich Optimal Transport Problem,'', PhD thesis, (2004).

[2]

N. Ahmad, H. K. Kim and R.J. McCann, Optimal transportation, topology and uniqueness,, Bull. Math. Sci., 1 (2011), 13. 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. doi: 10.1007/PL00004691.

[4]

L. Ambrosio, Lecture notes on optimal transport problems,, in, 1812 (2003), 1. 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. doi: 10.1007/s002110050002.

[7]

J.-P. Bourguignon, Ricci curvature and measures,, Japan. J. Math., 4 (2009), 27. 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.

[9]

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

[10]

L. A. Caffarelli, The regularity of mappings with a convex potential,, J. Amer. Math. Soc., 5 (1992), 99. 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. 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. 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. 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. 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. 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. 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. 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. 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), 65 (2013), 757. 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, (1976).

[21]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem,, Mem. Amer. Math. Soc., 137 (1999), 1. 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. 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. 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. 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. 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., 15 (2013), 1131. 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. 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. doi: 10.1002/cpa.20293.

[29]

A. Figalli, L. Rifford and C. Villani, Nearly round spheres look convex,, Amer. J. Math., 134 (2012), 109. 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. 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. doi: 10.1007/BF02392620.

[33]

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

[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.

[35]

F. R. Harvey and H. B. Lawson, Jr, Split special Lagrangian geometry,, Progress in Mathematics 297 (2012), 297 (2012), 43. 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. doi: 10.2307/3318478.

[37]

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

[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. 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. 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.

[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. doi: 10.1137/120865409.

[42]

T. C. Koopmans and M. Beckmann, Assignment problems and the location of economic activities,, Econometrica, 25 (1957), 53. 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. 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. 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. 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. doi: 10.1023/A:1008753021652.

[47]

J. Li, "Smooth Optimal Transportation on Hyperbolic Space,'', Master's thesis, (2009).

[48]

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

[49]

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