Abstract
The Monge–Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff (Lattice Theory. Revised Edition, 1948): give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff’s question culminate in Hestir and Williams’ necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure γ outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph/antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.
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Abdellaoui T., Heinich H.: Sur la distance de deux lois dans le cas vectoriel. C.R. Acad. Sci. Paris Sér. I Math. 319, 397–400 (1994)
Agrachev A., Lee P.W.Y.: Optimal transportation under nonholomic constraints. Trans. Am. Math. Soc. 361, 6019–6047 (2009)
Ahmad, N.: The geometry of shape recognition via a Monge–Kantorovich optimal transport problem. PhD thesis. Brown University (2004)
Ambrosio, L.: Lecture notes on optimal transport problems. In: Colli, P., Rodrigues, J.F. (eds.) Mathematical Aspects of Evolving Interfaces. Lecture Notes in Mathematics, vol. 1812, pp. 1–52. Springer, Berlin (2003)
Ambrosio L., Rigot S.: Optimal transportation in the Heisenberg group. J. Funct. Anal. 208, 261–301 (2004)
Ambrosio L., Kirchheim B., Pratelli A.: Existence of optimal transport maps for crystalline norms. Duke Math. J. 125, 207–241 (2004)
Ambrosio, L.A., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lecture Notes in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005)
Beneš, V., Štěpán, J.: The support of extremal probability measures with given marginals. In: Puri, M.L., Révész, P., Wertz, W. (eds.) Mathematical Statistics and Probability Theory, vol. A: Theoretical Aspects, pp. 33–41. D. Reidel Publishing Co., Dordrecht (1987)
Bernard P., Buffoni B.: The Monge problem for supercritical Mañé potentials on compact manifolds. Adv. Math. 207, 691–706 (2006)
Bernard P., Buffoni B.: Optimal mass transportation and Mather theory. J. Eur. Math. Soc. 9, 85–121 (2007)
Bertrand J.: Existence and uniqueness of optimal maps on Alexandrov spaces. Adv. Math. 219, 838–851 (2008)
Bianchini, S., Cavalletti, F.: The Monge problem for distance cost in geodesic spaces. Preprint at http://cvgmt.sns.it/papers/biacav09/Monge@problem.pdf
Birkhoff G.: Three observations on linear algebra. Univ. Nac. Tucumán. Revista A 5, 147–151 (1946)
Birkhoff, G.: Lattice Theory. Revised Edition, vol. 25 of Colloquium Publications. American Mathematical Society, New York (1948)
Bouchitté G., Buttazzo G., Seppecher P.: Shape optimization solutions via Monge–Kantorovich equation. C.R. Acad. Sci. Paris Sér. I 324, 1185–1191 (1997)
Bouchitté G., Gangbo W., Seppecher P.: Michell trusses and lines of principal action. Math. Models Methods Appl. Sci. 18, 1571–1603 (2008)
Brenier Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C.R. Acad. Sci. Paris Sér. I Math. 305, 805–808 (1987)
Brenier Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375–417 (1991)
Caffarelli, L.: Allocation maps with general cost functions. In: Marcellini, P., et al. (eds.) Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math., vol. 177, pp. 29–35. Dekker, New York (1996)
Caffarelli L.A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5, 99–104 (1992)
Caffarelli L.A.: Boundary regularity of maps with convex potentials—II. Ann. Math. (2) 144, 453–496 (1996)
Caffarelli L.A., Feldman M., McCann R.J.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Am. Math. Soc. 15, 1–26 (2002)
Caravenna, L.: An existence result for the Monge problem in Rn with norm cost functions. Preprint at http://cvgmt.sns.it/cgi/get.cgi/papers/cara/sel.partMonge.pdf
Carlier G.: A general existence result for the principal-agent problem with adverse selection. J. Math. Econ. 35, 129–150 (2001)
Carlier G., Ekeland I.: Structure of cities. J. Global Optim. 29, 371–376 (2004)
Champion, T., De Pascale, L.: The Monge problem in Rd. To appear in Duke Math. J. Preprint at http://cvgmt.sns.it/papers/chadep09/champion-depascale.pdf (2011)
Chiappori P.-A., McCann R.J., Nesheim L.: Hedonic price equilibria, stable matching and optimal transport: equivalence, topology and uniqueness. Econ. Theory 42, 317–354 (2010)
Cordero-Erausquin D.: Sur le transport de mesures périodiques. C.R. Acad. Sci. Paris Sér. I Math. 329, 199–202 (1999)
Cordero-Erausquin D., McCann R.J., Schmuckenschläger M.: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146, 219–257 (2001)
Cordero-Erausquin D., Nazaret B., Villani C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)
Cuesta-Albertos J.A., Matrán C.: Notes on the Wasserstein metric in Hilbert spaces. Ann. Probab. 17, 1264–1276 (1989)
Cuesta-Albertos J.A., Tuero-Díaz A.: A characterization for the solution of the Monge–Kantorovich mass transference problem. Stat. Probab. Lett. 16, 147–152 (1993)
Cullen M.J.P.: A Mathematical Theory of Large Scale Atmosphere/Ocean Flows. Imperial College Press, London (2006)
Cullen M.J.P., Purser R.J.: An extended Lagrangian model of semi-geostrophic frontogenesis. J. Atmos. Sci. 41, 1477–1497 (1984)
Cullen M.J.P., Purser R.J.: Properties of the Lagrangian semi-geostrophic equations. J. Atmos. Sci. 46, 2684–2697 (1989)
Delanoë P.: 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, 443–457 (1991)
Denny J.L.: The support of discrete extremal measures with given marginals. Mich. Math. J. 27, 59–64 (1980)
Dobrushin R.: Definition of a system of random variables by means of conditional distributions (Russian). Teor. Verojatnost. i Primenen. 15, 469–497 (1970)
Douglas R.D.: On extremal measures and subspace density. Mich. Math. J. 11, 243–246 (1964)
Dudley R.M.: Real Analysis and Probability. Revised reprint of the 1989 original. Cambridge University Press, Cambridge (2002)
Ekeland I.: Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types. Econ. Theory 42, 275–315 (2010)
Evans L.C., Gangbo W.: Differential equations methods for the Monge–Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137, 1–66 (1999)
Fathi A., Figalli A.: Optimal transportation on non-compact manifolds. Israel J. Math. 175, 1–59 (2010)
Feldman M., McCann R.J.: Monge’s transport problem on a Riemannian manifold. Trans. Am. Math. Soc. 354, 1667–1697 (2002)
Feldman M., McCann R.J.: Uniqueness and transport density in Monge’s transportation problem. Calc. Var. Partial Differ. Equ. 15, 81–113 (2002)
Figalli A.: Existence, uniqueness and regularity of optimal transport maps. SIAM J. Math. Anal. 39, 126–137 (2007)
Figalli A.: The Monge problem on non-compact manifolds. Rend. Sem. Mat. Univ. Padova 117, 147–166 (2007)
Figalli, A., Gigli, N.: Local semiconvexity of Kantorovich potentials on noncompact manifolds. To appear in ESAIM Control Optim. Calc. Var. (2011)
Figalli, A., Kim, Y.-H., McCann, R.J.: When is multidimensional screening a convex program? J. Econ. Theory 146, 454–478 (2011)
Figalli A., Rifford L.: Mass transportation on sub-Riemannian manifolds. Geom. Funct. Anal. 20, 124–159 (2010)
Figalli A., Maggi F., Pratelli A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182, 167–211 (2010)
Gangbo, W.: Habilitation thesis. Université de Metz, Metz (1995)
Gangbo W., McCann R.J.: Optimal maps in Monge’s mass transport problem. C.R. Acad. Sci. Paris Sér. I Math. 321, 1653–1658 (1995)
Gangbo W., McCann R.J.: The geometry of optimal transportation. Acta Math. 177, 113–161 (1996)
Gangbo W., McCann R.J.: Shape recognition via Wasserstein distance. Q. Appl. Math. 58, 705–737 (2000)
Gigli, N.: On the inverse implication of Brenier–McCann theorems and the structure of (P2(M),W2). Preprint at http://cvgmt.sns.it/papers/gigc/Inverse.pdf
Glimm T., Oliker V.: Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem. J. Math. Sci. 117, 4096–4108 (2003)
Hestir K., Williams S.C.: Supports of doubly stochastic measures. Bernoulli 1, 217–243 (1995)
Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)
Kantorovich L.: On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.) 37, 199–201 (1942)
Kantorovich L.: On a problem of Monge (In Russian). Uspekhi Math. Nauk. 3, 225–226 (1948)
Kim Y.-H., McCann R.J.: Continuity, curvature, and the general covariance of optimal transportation. J. Eur. Math. Soc. 12, 1009–1040 (2010)
Kim Y.-H., McCann R.J., Warren M.: Pseudo-Riemannian geometry calibrates optimal transportation. Math. Res. Lett. 17, 1183–1197 (2010)
Knott M., Smith C.S.: On the optimal mapping of distributions. J. Optim. Theory Appl. 43, 39–49 (1984)
Koopmans T.C.: Optimum utilization of the transportation system. Econometrica (Supplement) 17, 136–146 (1949)
Levin V.L.: Abstract cyclical monotonicity and Monge solutions for the general Monge–Kantorovich problem. Set-valued Anal. 7, 7–32 (1999)
Lindenstrauss J.: A remark on doubly stochastic measures. Am. Math. Monthly 72, 379–382 (1965)
Loeper G.: On the regularity of solutions of optimal transportation problems. Acta Math. 202, 241–283 (2009)
Losert V.: Counterexamples to some conjectures about doubly stochastic measures. Pac. J. Math. 99, 387–397 (1982)
Lott J.: Optimal transport and Perelman’s reduced volume. Calc. Var. Partial Differ. Equ. 36, 49–84 (2009)
Lott J., Villani C.: Ricci curvature for metric measure spaces via optimal transport. Ann. Math. (2) 169, 903–991 (2009)
Ma X.-N., Trudinger N., Wang X.-J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177, 151–183 (2005)
McCann, R.J.: A Convexity Theory for Interacting Gases and Equilibrium Crystals. PhD thesis. Princeton University, Princeton (1994)
McCann R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)
McCann R.J.: Equilibrium shapes for planar crystals in an external field. Comm. Math. Phys. 195, 699–723 (1998)
McCann R.J.: Exact solutions to the transportation problem on the line. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455, 1341–1380 (1999)
McCann R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11, 589–608 (2001)
McCann, R.J., Sosio, M.: Hölder continuity of optimal multivalued mappings. Preprint at http://www.math.toronto/mccann
McCann R.J., Topping P.: Ricci flow, entropy, and optimal transportation. Am. J. Math. 132, 711–730 (2010)
McCann, R.J., Pass, B., Warren, M.: Rectifiability of optimal transportation plans. Preprint at http://www.math.toronto/mccann
Monge, G.: Mémoire sur la théorie des déblais et de remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pp. 666–704 (1781)
Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equ. 26, 101–174 (2001)
Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)
Pass, B.: On the local structure of optimal measures in the multi-marginal optimal transportation problem. Preprint at http://arxiv.org/abs/1005.2162
Plakhov A.Yu.: Exact solutions of the one-dimensional Monge–Kantorovich problem (Russian). Mat. Sb. 195, 57–74 (2004)
Plakhov A.Yu.: Newton’s problem of the body of minimal averaged resistance (Russian). Mat. Sb. 195, 105–126 (2004)
Purser J., Cullen M.J.P.: J. Atmos. Sci. 44, 3449–3468 (1987)
Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems. Probab. Appl. Springer, New York (1998)
Rüschendorf L.: Optimal solutions of multivariate coupling problems. Appl. Math. (Warsaw) 23, 325–338 (1995)
Rüschendorf L.: On c-optimal random variables. Stat. Probab. Lett. 37, 267–270 (1996)
Rüschendorf L., Rachev S.T.: A characterization of random variables with minimum L2-distance. J. Multivar. Anal. 32, 48–54 (1990)
Seethoff T.L., Shiflett R.C.: Doubly stochastic measures with prescribed support. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41, 283–288 (1978)
Smith C., Knott M.: On the optimal transportation of distributions. J. Optim. Theory Appl. 52, 323–329 (1987)
Smith C., Knott M.: On Hoeffding-Fréchet bounds and cyclic monotone relations. J. Multivar. Anal. 40, 328–334 (1992)
Sturm K.-T.: On the geometry of metric measure spaces, I and II. Acta Math. 196, 65–177 (2006)
Sudakov V.N.: Geometric problems in the theory of infinite-dimensional probability distributions. Proc. Steklov Inst. Math. 141, 1–178 (1979)
Tanaka H.: An inequality for a functional of probability distributions and its application to Kac’s one-dimensional model of a Maxwellian gas. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27, 47–52 (1973)
Trudinger N.S.: Isoperimetric inequalities for quermassintegrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 11, 411–425 (1994)
Trudinger N.S., Wang X.-J.: On the Monge mass transfer problem. Calc. Var. Partial Differ. Equ. 13, 19–31 (2001)
Trudinger N.S., Wang X.-J.: On the second boundary value problem for Monge–Ampère type equations and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8, 1–32 (2009)
Uckelmann L.: Optimal couplings between onedimensional distributions. In: Benes, V., Stepan, J. (eds) Distributions with given marginals and moment problems, pp. 261–273. Kluwer Academic Publishers, Dordrecht (1997)
Urbas J.: On the second boundary value problem for equations of Monge–Ampère type. J. Reine Angew. Math. 487, 115–124 (1997)
Villani, C.: Cours d’Intégration et Analyse de Fourier. http://www.umpa.ens-lyon.fr/~cvillani/Cours/iaf-2006.html(2006)
Villani, C.: Optimal Transport. Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, New York (2009)
von Neumann, J.: A certain zero-sum two-person game equivalent to the optimal assignment problem. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the theory of games, vol. 2, pp. 5–12. Princeton University Press, Princeton, NJ (1953)
Wang X.-J.: On the design of a reflector antenna. Inverse Problems 12, 351–375 (1996)
Wang X.-J.: On the design of a reflector antenna II. Calc. Var. Partial Differ. Equ. 20, 329–341 (2004)
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Communicated by N. Trudinger.
Formerly titled Extremal doubly stochastic measures and optimal transportation.
It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) and Mathematical Sciences Research Institute in Berkeley (2005) for stimulating these developments by bringing us together. The authors are pleased to acknowledge the support of Natural Sciences and Engineering Research Council of Canada Grants 217006-03 and −08 and United States National Science Foundation Grant DMS-0354729. © 2009 by the authors.
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Ahmad, N., Kim, H.K. & McCann, R.J. Optimal transportation, topology and uniqueness. Bull. Math. Sci. 1, 13–32 (2011). https://doi.org/10.1007/s13373-011-0002-7
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DOI: https://doi.org/10.1007/s13373-011-0002-7