Existence and uniqueness of monotone measure-preserving maps

[1] Y. Brenier, <i>Décomposition polaire et réarrangement monotone des champs de vecteurs</i>, C. R. Acad. Sci. Paris Sér. I Math. <b>305</b> (1987), no. 19, 805–808.

[2] Y. Brenier, <i>Polar factorization and monotone rearrangement of vector-valued functions</i>, Comm. Pure Appl. Math. <b>44</b> (1991), no. 4, 375–417.

[3] R. T. Rockafellar, <i>Characterization of the subdifferentials of convex functions</i>, Pacific J. Math. <b>17</b> (1966), 497–510.

[4] R. D. Anderson and V. L. Klee, Jr., <i>Convex functions and upper semi-continuous collections</i>, Duke Math. J. <b>19</b> (1952), 349–357.

[5] A. D. Alexandroff, <i>Existence and uniqueness of a convex surface with a given integral curvature</i>, C. R. (Doklady) Acad. Sci. URSS (N.S.) <b>35</b> (1942), 131–134.

[6] D. C. Dowson and B. V. Landau, <i>The Fréchet distance between multivariate normal distributions</i>, J. Multivariate Anal. <b>12</b> (1982), no. 3, 450–455.

[7] M. Knott and C. S. Smith, <i>On the optimal mapping of distributions</i>, J. Optim. Theory Appl. <b>43</b> (1984), no. 1, 39–49.

[8] L. Caffarelli, <i>Boundary regularity of maps with convex potentials</i>, Comm. Pure Appl. Math. <b>45</b> (1992), no. 9, 1141–1151.

[9] W. Gangbo, <i>An elementary proof of the polar factorization of vector-valued functions</i>, Arch. Rational Mech. Anal. <b>128</b> (1994), no. 4, 381–399.

[10] L. Caffarelli, <i>The regularity of mappings with a convex potential</i>, J. Amer. Math. Soc. <b>5</b> (1992), no. 1, 99–104.

[11] L. Caffarelli, <i>Boundary regularity of maps with convex potentials–$2$</i>, preprint.

[12] R. J. McCann, <i>A Convexity Theory for Interacting Gases and Equilibrium Crystals</i>, Ph.D. thesis, Princeton University, 1994.

[13] R. J. McCann, <i>A convexity principle for attracting gases</i>, to appear in Adv. Math.

[14] R. T. Rockafellar, <i>Convex Analysis</i>, Princeton University Press, Princeton, 1972.

[15] S. T. Rachev, <i>The Monge-Kantorovich mass transference problem and its stochastic applications</i>, Theory Probab. Appl. <b>29</b> (1984), 647–676.

[16] C. Smith and M. Knott, <i>On Hoeffding-Fréchet bounds and cyclic monotone relations</i>, J. Multivariate Anal. <b>40</b> (1992), no. 2, 328–334.

[17] L. Rüschendorf, <i>Fréchet-bounds and their applications</i>, Advances in probability distributions with given marginals (Rome, 1990) eds. G. Dall'Agilo, S. Kotz, and G. Salietti, Math. Appl., vol. 67, Kluwer Acad. Publ., Dordrecht, 1991, pp. 151–187.

[18] T. Abdellaoui and H. Heinich, <i>Sur la distance de deux lois dans le cas vectoriel</i>, C. R. Acad. Sci. Paris Sér. I Math. <b>319</b> (1994), no. 4, 397–400.

[19] L. Rüschendorf and S. T. Rachev, <i>A characterization of random variables with minimum $L\sp 2$-distance</i>, J. Multivariate Anal. <b>32</b> (1990), no. 1, 48–54.

[20] J. A. Cuesta-Albertos, C. Matrán, and A. Tuero-Díaz, <i>Optimal maps for the $L^2$-Wasserstein distance</i>, preprint.