×

A glimpse into the differential topology and geometry of optimal transport. (English) Zbl 1275.49086

Summary: 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
58E30 Variational principles in infinite-dimensional spaces
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
90B06 Transportation, logistics and supply chain management
90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)
91B32 Resource and cost allocation (including fair division, apportionment, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] N. Ahmad, “The Geometry of Shape Recognition Via a Monge-Kantorovich Optimal Transport Problem,”, PhD thesis (2004)
[2] N. Ahmad, <em>Optimal transportation, topology and uniqueness,</em>, Bull. Math. Sci., 1, 13 (2011) · Zbl 1255.49075 · doi:10.1007/s13373-011-0002-7
[3] G. Alberti, <em>A geometrical approach to monotone functions in \(\mathbbR^n\)</em>,, Math. Z., 230, 259 (1999) · Zbl 0934.49025 · doi:10.1007/PL00004691
[4] L. Ambrosio, <em>Lecture notes on optimal transport problems,</em>, in, 1812, 1 (2003) · Zbl 1047.35001 · doi:10.1007/978-3-540-39189-0_1
[5] L. A. Ambrosio, <em>A user’s guide to optimal transport,</em>, Preprint. · doi:10.1007/978-3-642-32160-3_1
[6] J.-D. Benamou, <em>A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,</em>, Numer. Math., 84, 375 (2000) · Zbl 0968.76069 · doi:10.1007/s002110050002
[7] J.-P. Bourguignon, <em>Ricci curvature and measures,</em>, Japan. J. Math., 4, 27 (2009) · Zbl 1181.58026 · doi:10.1007/s11537-009-0855-7
[8] Y. Brenier, <em>Décomposition polaire et réarrangement monotone des champs de vecteurs,</em> (French) [Polar decomposition and monotone rearrangement of vector fields], C.R. Acad. Sci. Paris Sér. I Math., 305, 805 (1987) · Zbl 0652.26017
[9] Y. Brenier, <em>Polar factorization and monotone rearrangement of vector-valued functions,</em>, Comm. Pure Appl. Math., 44, 375 (1991) · Zbl 0738.46011 · doi:10.1002/cpa.3160440402
[10] L. A. Caffarelli, <em>The regularity of mappings with a convex potential,</em>, J. Amer. Math. Soc., 5, 99 (1992) · Zbl 0753.35031 · doi:10.1090/S0894-0347-1992-1124980-8
[11] L. A. Caffarelli, <em>Boundary regularity of maps with convex potentials - II</em>,, Ann. of Math. (2), 144, 453 (1996) · Zbl 0916.35016 · doi:10.2307/2118564
[12] L. A. Caffarelli, <em>Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs</em>,, J. Amer. Math. Soc., 15, 1 (2002) · Zbl 1053.49032 · doi:10.1090/S0894-0347-01-00376-9
[13] T. Champion, <em>The Monge problem in \(\mathbbR^d\)</em>,, Duke Math. J., 157, 551 (2011) · Zbl 1232.49050 · doi:10.1215/00127094-1272939
[14] P.-A. Chiappori, <em>Hedonic price equilibria, stable matching and optimal transport: Equivalence, topology and uniqueness,</em>, Econom. Theory, 42, 317 (2010) · Zbl 1183.91056 · doi:10.1007/s00199-009-0455-z
[15] D. Cordero-Erausquin, <em>A Riemannian interpolation inequality à la Borell, Brascamp and Lieb</em>,, Invent. Math., 146, 219 (2001) · Zbl 1026.58018 · doi:10.1007/s002220100160
[16] M. J. P. Cullen, <em>An extended Lagrangian model of semi-geostrophic frontogenesis,</em>, J. Atmos. Sci., 41, 1477 (1984) · doi:10.1175/1520-0469(1984)041<1477:AELTOS>2.0.CO;2
[17] P. Delanoë, <em>Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator</em>,, Ann. Inst. H. Poincarè Anal. Non Linèaire, 8, 443 (1991) · Zbl 0778.35037 · doi:10.1016/j.anihpc.2007.03.001
[18] P. Delanoë, <em>Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds,</em>, J. Reine Angew. Math., 646, 65 (2010) · Zbl 1200.58025 · doi:10.1515/CRELLE.2010.066
[19] P. Delanoë, <em>Positively curved Riemannian locally symmetric spaces are positively square distance curved</em>,, Canad. J. Math. 65 (2013), 65, 757 (2013) · Zbl 1312.53073 · 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) · Zbl 0355.60004
[21] L. C. Evans, <em>Differential equations methods for the Monge-Kantorovich mass transfer problem,</em>, Mem. Amer. Math. Soc., 137, 1 (1999) · Zbl 0920.49004 · doi:10.1090/memo/0653
[22] M. Feldman, <em>Uniqueness and transport density in Monge’s transportation problem</em>,, Calc. Var. Partial Differential Equations, 15, 81 (2002) · Zbl 1003.49031 · doi:10.1007/s005260100119
[23] A. Figalli, <em>Regularity properties of optimal maps between nonconvex domains in the plane,</em>, Comm. Partial Differential Equations, 35, 465 (2010) · Zbl 1193.35086 · doi:10.1080/03605300903307673
[24] A. Figalli, <em>Partial regularity of Brenier solutions of the Monge-Ampère equation</em>,, Discrete Contin. Dyn. Syst., 28, 559 (2010) · Zbl 1193.35087 · doi:10.3934/dcds.2010.28.559
[25] A. Figalli, <em>Hölder continuity and injectivity of optimal maps</em>,, Arch. Rational Mech. Anal., 209, 747 (2013) · Zbl 1281.49037 · doi:10.1007/s00205-013-0629-5
[26] A. Figalli, <em>Regularity of optimal transport maps on multiple products of spheres</em>,, J. Euro. Math. Soc., 15, 1131 (2013) · Zbl 1268.49053 · doi:10.4171/JEMS/388
[27] A. Figalli, <em>When is multidimensional screening a convex program?</em>,, J. Econom Theory, 146, 454 (2011) · Zbl 1282.90085 · doi:10.1016/j.jet.2010.11.006
[28] A. Figalli, <em>Continuity of optimal transport maps on small deformations of \(\mathbbS^2\),</em>, Comm. Pure Appl. Math., 62, 1670 (2009) · Zbl 1175.49040 · doi:10.1002/cpa.20293
[29] A. Figalli, <em>Nearly round spheres look convex,</em>, Amer. J. Math., 134, 109 (2012) · Zbl 1241.53031 · doi:10.1353/ajm.2012.0000
[30] L. Forzani, <em>Properties of the solutions to the Monge-Ampère equation,</em>, Nonlinear Anal., 57, 815 (2004) · Zbl 1137.35361 · doi:10.1016/j.na.2004.03.019
[31] W. Gangbo., “Habilitation Thesis,”, Université de Metz (1995)
[32] W. Gangbo, <em>The geometry of optimal transportation</em>,, Acta Math., 177, 113 (1996) · Zbl 0887.49017 · doi:10.1007/BF02392620
[33] W. Gangbo, <em>Shape recognition via Wasserstein distance</em>,, Quart. Appl. Math., 58, 705 (2000) · Zbl 1039.49038
[34] N. Gigli, <em>On the inverse implication of Brenier-McCann theorems and the structure of \((P_2(M),W_2)\)</em>,, Methods Appl. Anal., 18, 127 (2011) · Zbl 1284.49050
[35] F. R. Harvey, <em>Split special Lagrangian geometry</em>,, Progress in Mathematics 297 (2012), 297, 43 (2012) · Zbl 1250.53048 · doi:10.1007/978-3-0348-0257-4_3
[36] K. Hestir, <em>Supports of doubly stochastic measures,</em>, Bernoulli, 1, 217 (1995) · Zbl 0844.60002 · doi:10.2307/3318478
[37] L. Kantorovich, <em>On the translocation of masses,</em>, C. R. (Doklady) Acad. Sci. URSS (N. S.), 37, 199 (1942) · Zbl 0061.09705
[38] Y.-H. Kim, <em>Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)</em>,, J. Reine Angew. Math., 664, 1 (2012) · Zbl 1239.53049 · doi:10.1515/CRELLE.2011.105
[39] Y.-H. Kim, <em>Continuity, curvature, and the general covariance of optimal transportation,</em>, J. Eur. Math. Soc. (JEMS), 12, 1009 (2010) · Zbl 1191.49046 · doi:10.4171/JEMS/221
[40] Y.-H. Kim, <em>Pseudo-Riemannian geometry calibrates optimal transportation</em>,, Math. Res. Lett., 17, 1183 (2010) · Zbl 1222.49059
[41] J. Kitagawa, <em>Regularity for the optimal transportation problem with Euclidean distance squared cost on the embedded sphere,</em>, SIAM J. Math. Anal., 44, 2871 (2012) · Zbl 1273.35275 · doi:10.1137/120865409
[42] T. C. Koopmans, <em>Assignment problems and the location of economic activities,</em>, Econometrica, 25, 53 (1957) · Zbl 0098.12203 · doi:10.2307/1907742
[43] P. W. Y. Lee, <em>New computable necessary conditions for the regularity theory of optimal transportation</em>,, SIAM J. Math. Anal., 42, 3054 (2010) · Zbl 1234.49039 · doi:10.1137/100797722
[44] P. W. Y. Lee, <em>New examples on spaces of negative sectional curvature satisfying Ma-Trudinger-Wang conditions</em>,, SIAM J. Math. Anal., 44, 61 (2012) · Zbl 1243.58007 · doi:10.1137/110820543
[45] P. W. Y. Lee, <em>The Ma-Trudinger-Wang curvature for natural mechanical actions</em>,, Calc. Var. Partial Differential Equations, 41, 285 (2011) · Zbl 1220.37045 · doi:10.1007/s00526-010-0362-y
[46] V. L. Levin, <em>Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem,</em>, Set-valued Anal., 7, 7 (1999) · Zbl 0934.54013 · doi:10.1023/A:1008753021652
[47] J. Li, “Smooth Optimal Transportation on Hyperbolic Space,”, Master’s thesis (2009)
[48] J. Liu, <em>Hölder regularity of optimal mappings in optimal transportation,</em>, Calc Var. Partial Differential Equations, 34, 435 (2009) · Zbl 1166.35331 · doi:10.1007/s00526-008-0190-5
[49] J. Liu, <em>Interior \(C^{2,\alpha}\) regularity for potential functions in optimal transportation,</em>, Comm. Partial Differential Equations, 35, 165 (2010) · Zbl 1189.35142 · doi:10.1080/03605300903236609
[50] G. Loeper, <em>On the regularity of solutions of optimal transportation problems,</em>, Acta Math., 202, 241 (2009) · Zbl 1219.49038 · doi:10.1007/s11511-009-0037-8
[51] G. Loeper, <em>Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna</em>,, Arch. Ration. Mech. Anal., 199, 269 (2011) · Zbl 1231.35280 · doi:10.1007/s00205-010-0330-x
[52] G. Loeper, <em>Regularity of optimal transport in curved geometry: The non-focal case,</em>, Duke Math. J., 151, 431 (2010) · Zbl 1192.53041 · doi:10.1215/00127094-2010-003
[53] G. G. Lorentz, <em>An inequality for rearrangements,</em>, Amer. Math. Monthly, 60, 176 (1953) · Zbl 0050.28201 · doi:10.2307/2307574
[54] J. Lott, <em>Ricci curvature for metric measure spaces via optimal transport,</em>, Annals Math. (2), 169, 903 (2009) · Zbl 1178.53038 · doi:10.4007/annals.2009.169.903
[55] X.-N. Ma, <em>Regularity of potential functions of the optimal transportation problem,</em>, Arch. Rational Mech. Anal., 177, 151 (2005) · Zbl 1072.49035 · doi:10.1007/s00205-005-0362-9
[56] R. J. McCann, <em>Existence and uniqueness of monotone measure-preserving maps,</em>, Duke Math. J., 80, 309 (1995) · Zbl 0873.28009 · doi:10.1215/S0012-7094-95-08013-2
[57] R. J. McCann, <em>A convexity principle for interacting gases,</em>, Adv. Math., 128, 153 (1997) · Zbl 0901.49012 · doi:10.1006/aima.1997.1634
[58] R. J. McCann, <em>Exact solutions to the transportation problem on the line,</em>, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455, 1341 (1999) · Zbl 0947.90010 · doi:10.1098/rspa.1999.0364
[59] R. J. McCann, <em>Polar factorization of maps on Riemannian manifolds,</em>, Geom. Funct. Anal., 11, 589 (2001) · Zbl 1011.58009 · doi:10.1007/PL00001679
[60] R. J. McCann, <em>Five lectures on optimal transportation: geometry, regularity, and applications,</em>, in, 145 (2013) · Zbl 1271.49034
[61] R. J. McCann, <em>Hölder continuity of optimal multivalued mappings</em>,, SIAM J. Math. Anal., 43, 1855 (2011) · Zbl 1229.49040 · doi:10.1137/100802670
[62] R. J. McCann, <em>Rectifiability of optimal transportation plans,</em>, Canad. J. Math, 64, 924 (2012) · Zbl 1248.49060 · doi:10.4153/CJM-2011-080-6
[63] G. J. Minty, <em>Monotone (nonlinear) operators in Hilbert space</em>,, Duke Math. J., 29, 341 (1962) · Zbl 0111.31202 · doi:10.1215/S0012-7094-62-02933-2
[64] J. A. Mirrlees, <em>An exploration in the theory of optimum income taxation,</em>, Rev. Econom. Stud., 38, 175 (1971) · Zbl 0222.90028
[65] G. Monge, <em>Mémoire sur la théorie des déblais et de remblais,</em>, in, 666 (1781)
[66] F. Otto, <em>The geometry of dissipative evolution equations: The porous medium equation,</em>, Comm. Partial Differential Equations, 26, 101 (2001) · Zbl 0984.35089 · doi:10.1081/PDE-100002243
[67] F. Otto, <em>Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality</em>,, J. Funct. Anal., 173, 361 (2000) · Zbl 0985.58019 · doi:10.1006/jfan.1999.3557
[68] B. Pass, <em>On the local structure of optimal measures in the multi-marginal optimal transportation problem</em>,, Calc. Var. Partial Differential Equations, 43, 529 (2012) · Zbl 1231.49037 · doi:10.1007/s00526-011-0421-z
[69] A. Pratelli, <em>On the equality between Monge’s infimum and Kantorovich’s minimum in optimal mass transportation</em>,, Ann. Inst. H. Poincaré Probab. Statist., 43, 1 (2007) · Zbl 1121.49036 · doi:10.1016/j.anihpb.2005.12.001
[70] A. Pratelli, <em>On the sufficiency of c-cyclical monotonicity for optimality of transport plans,</em>, Math. Z., 258, 677 (2008) · Zbl 1293.49110 · doi:10.1007/s00209-007-0191-7
[71] S. T. Rachev, “Mass Transportation Problems,”, Vol. I. Theory. Probability and its Applications (New York). Springer-Verlag (1998) · Zbl 0990.60500
[72] M.-K. von Renesse, <em>Transport inequalities, gradient estimates, entropy and Ricci curvature</em>,, Comm. Pure Appl. Math., 58, 923 (2005) · Zbl 1078.53028 · doi:10.1002/cpa.20060
[73] J.-C. Rochet, <em>A necessary and sufficient condition for rationalizability in a quasi-linear context,</em>, J. Math. Econom., 16, 191 (1987) · Zbl 0628.90003 · doi:10.1016/0304-4068(87)90007-3
[74] R. T. Rockafellar, <em>Characterization of the subdifferentials of convex functions,</em>, Pacific J. Math., 17, 497 (1966) · Zbl 0145.15901 · doi:10.2140/pjm.1966.17.497
[75] L. Rüschendorf, <em>A characterization of random variables with minimum \(L^2\)-distance,</em>, J. Multivariate Anal., 32, 48 (1990) · Zbl 0688.62034 · doi:10.1016/0047-259X(90)90070-X
[76] W. Schachermayer, <em>Characterization of optimal transport plans for the Monge-Kantorovich problem</em>,, Proc. Amer. Math. Soc., 137, 519 (2009) · Zbl 1165.49015 · doi:10.1090/S0002-9939-08-09419-7
[77] C. Smith, <em>Note on the optimal transportation of distributions,</em>, J. Optim. Theory Appl., 52, 323 (1987) · Zbl 0586.49005 · doi:10.1007/BF00941290
[78] M. Spence, <em>Job market signaling,</em>, Quarterly J. Econom., 87, 355 (1973) · doi:10.2307/1882010
[79] K.-T. Sturm, <em>On the geometry of metric measure spaces, I. </em>,, Acta Math., 196, 65 (2006) · Zbl 1105.53035 · doi:10.1007/s11511-006-0002-8
[80] N. S. Trudinger, <em>On the Monge mass transfer problem</em>,, Calc. Var. Paritial Differential Equations, 13, 19 (2001) · Zbl 1010.49030 · doi:10.1007/PL00009922
[81] N. S. Trudinger, <em>On the second boundary value problem for Monge-Ampère type equations and optimal transportation</em>,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8, 143 (2009) · Zbl 1182.35134
[82] J. Urbas, <em>On the second boundary value problem for equations of Monge-Ampère type</em>,, J. Reine Angew. Math., 487, 115 (1997) · Zbl 0880.35031 · doi:10.1515/crll.1997.487.115
[83] C. Villani, “Topics in Optimal Transportation,”, Graduate Studies in Mathematics (2003) · Zbl 1013.00028 · doi:10.1007/b12016
[84] C. Villani, “Optimal Transport. Old and New,”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (2009) · Zbl 1156.53003 · doi:10.1007/978-3-540-71050-9
[85] X.-J. Wang, <em>On the design of a reflector antenna</em>,, Inverse Problems, 12, 351 (1996) · Zbl 0858.35142 · doi:10.1088/0266-5611/12/3/013
[86] X.-J. Wang, <em>On the design of a reflector antenna II</em>,, Calc. Var. Partial Differential Equations, 20, 329 (2004) · Zbl 1065.78013 · doi:10.1007/s00526-003-0239-4
[87] Y. Yu, <em>Singular set of a convex potential in two dimensions,</em>, Comm. Partial Differential Equations, 32, 1883 (2007) · Zbl 1138.35019 · doi:10.1080/03605300701318757
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.