## Existence and uniqueness of monotone measure-preserving maps.(English)Zbl 0873.28009

The author proves existence and uniqueness of monotone measure-preserving maps. Furthermore, he gives an example in which existence and uniqueness fail, and an appendix containing a version of the implicit function theorem used in the proof of uniqueness; it applies to functions which, though not continuously differentiable, are differences of two convex functions.

### MSC:

 28D05 Measure-preserving transformations

### Keywords:

existence; uniqueness; monotone measure-preserving maps
Full Text:

### References:

 [1] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs , C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 19, 805-808. · Zbl 0652.26017 [2] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions , Comm. Pure Appl. Math. 44 (1991), no. 4, 375-417. · Zbl 0738.46011 [3] R. T. Rockafellar, Characterization of the subdifferentials of convex functions , Pacific J. Math. 17 (1966), 497-510. · Zbl 0145.15901 [4] R. D. Anderson and V. L. Klee, Jr., Convex functions and upper semi-continuous collections , Duke Math. J. 19 (1952), 349-357. · Zbl 0047.15702 [5] A. D. Alexandroff, Existence and uniqueness of a convex surface with a given integral curvature , C. R. (Doklady) Acad. Sci. URSS (N.S.) 35 (1942), 131-134. · Zbl 0061.37604 [6] D. C. Dowson and B. V. Landau, The Fréchet distance between multivariate normal distributions , J. Multivariate Anal. 12 (1982), no. 3, 450-455. · Zbl 0501.62038 [7] M. Knott and C. S. Smith, On the optimal mapping of distributions , J. Optim. Theory Appl. 43 (1984), no. 1, 39-49. · Zbl 0519.60010 [8] L. Caffarelli, Boundary regularity of maps with convex potentials , Comm. Pure Appl. Math. 45 (1992), no. 9, 1141-1151. · Zbl 0778.35015 [9] W. Gangbo, An elementary proof of the polar factorization of vector-valued functions , Arch. Rational Mech. Anal. 128 (1994), no. 4, 381-399. · Zbl 0828.57021 [10] L. Caffarelli, The regularity of mappings with a convex potential , J. Amer. Math. Soc. 5 (1992), no. 1, 99-104. JSTOR: · Zbl 0753.35031 [11] L. Caffarelli, Boundary regularity of maps with convex potentials-$$2$$ , · Zbl 0778.35015 [12] R. J. McCann, A Convexity Theory for Interacting Gases and Equilibrium Crystals , Ph.D. thesis, Princeton University, 1994. [13] R. J. McCann, A convexity principle for attracting gases , to appear in Adv. Math. · Zbl 0901.49012 [14] R. T. Rockafellar, Convex Analysis , Princeton University Press, Princeton, 1972. · Zbl 0224.49003 [15] S. T. Rachev, The Monge-Kantorovich mass transference problem and its stochastic applications , Theory Probab. Appl. 29 (1984), 647-676. · Zbl 0581.60010 [16] C. Smith and M. Knott, On Hoeffding-Fréchet bounds and cyclic monotone relations , J. Multivariate Anal. 40 (1992), no. 2, 328-334. · Zbl 0745.62055 [17] L. Rüschendorf, Fréchet-bounds and their applications , 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. · Zbl 0744.60005 [18] T. Abdellaoui and H. Heinich, Sur la distance de deux lois dans le cas vectoriel , C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 397-400. · Zbl 0808.60008 [19] L. Rüschendorf and S. T. Rachev, A characterization of random variables with minimum $$L^ 2$$-distance , J. Multivariate Anal. 32 (1990), no. 1, 48-54. · Zbl 0688.62034 [20] J. A. Cuesta-Albertos, C. Matrán, and A. Tuero-Díaz, Optimal maps for the $$L^2$$-Wasserstein distance ,
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