The geometry of optimal transportation

Abdellaoui, T., Détermination d'un couple optimal du problème de Monge-Kantorovitch.C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 981–984.

Abdellaoui, T. &Heinich, H., Sur la distance de deux lois dans le cas vectoriel.C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 397–400.

Alberti, G., On the structure of singular sets of convex functions.Calc. Var. Partial Differential Equations, 2 (1994), 17–27.

Aleksandrov, A. D., Existence and uniqueness of a convex surface with a given integral curvature.C. R. (Doklady) Acad. Sci. URSS (N.S.), 35 (1942), 131–134.

Appell, P., Memoire sur les déblais et les remblais des systèmes continues ou discontinues.Mémoires présentés par divers Savants à l'Académie des Sciences de l'Institut de France, Paris, I. N., 29 (1887), 1–208.

Balder, E. J., An extension of duality-stability relations to non-convex optimization problems.SIAM J. Control Optim. 15 (1977), 329–343.

Brenier, Y., Decomposition polaire et réarrangement monotone des champs de vecteurs.C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 805–808.

—, Polar factorization and monotone rearrangement of vector-valued functions.Comm. Pure Appl. Math., 44 (1991), 375–417.

Caffarelli, L., The regularity of mappings with a convex potential.J. Amer. Math. Soc., 5 (1992), 99–104.

—, Allocation maps with general cost functions, inPartial Differential Equations and Applications (P. Marcellini, G. Talenti and E. Vesintini, eds.), pp. 29–35. Lecture Notes in Pure and Appl. Math., 177. Dekker, New York, 1996.

Cuesta-Albertos, J. A. &Matrán, C., Notes on the Wasserstein metric in Hilbert spaces.Ann. Probab., 17 (1989), 1264–1276.

Cuesta-Albertos, J. A., Matrán, C. & Tuero-Díaz, A., Properties of the optimal maps for theL 2-Monge-Kantorovich transportation problem. Preprint.

Cuesta-Albertos, J. A., Rüschendorf, L. &Tuero-Díaz, A., Optimal coupling of multivariate distributions and stochastic processes.J. Multivariate Anal., 46 (1993), 335–361.

Cuesta-Albertos, J. A. &Tuero-Díaz, A., A characterization for the solution of the Monge-Kantorovich mass transference problem.Statist. Probab. Lett., 16 (1993), 147–152.

Dall'Aglio, G., Sugli estremi dei momenti delle funzioni di ripartizione-doppia.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 10 (1956), 35–74.

Darboux, G., Prix Bordin (géométrie).C. R. Acad. Sci. Paris, 101 (1885), 1312–1316.

Douglis, A., Solutions in the large for multi-dimensional non-linear partial differential equations of first order.Ann. Inst. Fourier (Grenoble), 15:2 (1965), 1–35.

Evans, L. C. & Gangbo, W., Differential equations methods for the Monge-Kantorovich mass transfer problem. Preprint.

Fréchet, M., Sur la distance de deux lois de probabilité.C. R. Acad. Sci. Paris, 244 (1957), 689–692.

Gangbo, W. &McCann, R. J., Optimal maps in Monge's mass transport problem.C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1653–1658.

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

—, On a problem of Monge.Uspekhi Mat. Nauk. 3 (1948), 225–226 (in Russian).

Katz, B. S. (editor),Nobel Laureates in Economic Sciences: A Biographical Dictionary. Garland Publishing Inc., New York, 1989.

Kellerer, H. G., Duality theorems for marginal problems.Z. Wahrsch. Verw. Gebiete, 67 (1984), 399–432.

Knott, M. &Smith, C. S., On the optimal mapping of distributions.J. Optim. Theory Appl., 43 (1984), 39–49.

Levin, V. L., General Monge-Kantorovich problem and its applications in measure theory and mathematical economics, inFunctional Analysis, Optimization, and Mathematical Economics (L. J. Leifman, ed.), pp. 141–176. Oxford Univ. Press, New York, 1990.

McCann, R. J., Existence and uniqueness of monotone measure-preserving maps.Duke Math. J., 80 (1995), 309–323.

McCann, R. J., A convexity principle for interacting gases. To appear inAdv. in Math.

McCann, R. J., Exact solutions to the transportation problem on the line. To appear.

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, 1781, pp. 666–704.

Rachev, S. T., The Monge-Kantorovich mass transference problem and its stochastic applications.Theory Probab. Appl., 29 (1984), 647–676.

Rockafellar, R. T., Characterization of the subdifferentials of convex functions.Pacific J. Math., 17 (1966), 497–510.

—,Convex Analysis. Princeton Univ. Press, Princeton, NJ, 1972.

Rüschendorf, L., Bounds for distributions with multivariate marginals, inStochastic Orders and Decision Under Risk (K. Mosler and M. Scarsini, eds.), pp. 285–310. IMS Lecture Notes-Monograph Series. Inst. of Math. Statist, Hayward, CA, 1991.

—, Frechet bounds and their applications, inAdvances in Probability Distributions with Given Marginals (G. Dall'Aglio et al., eds.), pp. 151–187. Math. Appl., 67. Kluwer Acad. Publ., Dordrecht, 1991.

—, Optimal solutions of multivariate coupling problems.Appl. Math. (Warszaw), 23 (1995), 325–338.

—, Onc-optimal random variables.Statist. Probab. Lett., 27 (1996), 267–270.

Rüschendorf, L. &Rachev, S. T., A characterization of random variables with minimumL 2-distance.J. Multivariate Anal., 32 (1990), 48–54.

Schneider, R.,Convex Bodies: The Brunn-Minkowski Theory. Cambridge Univ. Press, Cambridge, 1993.

Smith, C. &Knott, M., On the optimal transportation of distributions.J. Optim. Theory Appl., 52 (1987), 323–329.

—, On Hoeffding-Fréchet bounds and cyclic monotone relations.J. Multivariate Anal., 40 (1992), 328–334.

Sudakov, V. N., Geometric problems in the theory of infinite-dimensional probability distributions.Proc. Steklov Inst. Math., 141 (1979), 1–178.

Zajíĉek, L., On the differentiation of convex functions in finite and infinite dimensional spaces.Czechoslovak Math. J. 29 (104) (1979), 340–348.