Abstract
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and d L is a geodesic Borel distance which makes (X, d L ) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by d L . It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting d L -cyclical monotonicity is not sufficient for optimality.
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References
Ambrosio L., Kirchheim B.: Currents in metric spaces. Acta Mathematica 185(1), 1–80 (2000)
Ambrosio L., Kirchheim B., Pratelli A.: Existence of optimal transport maps for crystalline norms. Duke Math. J. 125(2), 207–241 (2004)
Bianchini S., Caravenna L.: On the extremality, uniqueness and optimality of transference plans. Bull. Inst. Math. Acad. Sin. (N.S.) 4(4), 353–454 (2009)
Bianchini S., Gloyer M.: The Euler-Lagrange equation for a singular variational problem. Math. Z. 265(4), 889–923 (2009)
Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate studies in mathematics. Providence, RI: Amer. Math. Soc., 2001
Caffarelli L., Feldman M., McCann R.J.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Amer. Math. Soc. 15, 1–26 (2002)
Caravenna, L.: An existence result for the Monge problem in \({\mathbb{R}^n}\) with norm cost function. Preprint, 2009, available at http://cvgnt.sns.it/paper-11303
Champion T., De Pascale L.: The Monge problem in \({\mathbb{R}^d}\). Duke Math. J. 157(3), 551–572 (2011)
Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem, Current Developments in Mathematics, Boston, MA: Dul Press, 1999, pp. 65–126
Feldman M., McCann R.: Monge’s transport problem on a Riemannian manifold. Trans. Amer. Math. Soc. 354, 1667–1697 (2002)
Fremlin, D.H.: Measure Theory, Volume 4. Torres Fremlin, 2002, available at http://www.essex.ac.vk/maths/people/termlin/mty.2006/index.html
Larman D.G.: A compact set of disjoint line segments in \({\mathbb{R}^{3}}\) whose end set has positive measure. Mathematika 18, 112–125 (1971)
Lott J., Villani C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. 169(3), 903–991 (2009)
Ohta S.-I.: On the measure contraction property of metric measure spaces. Comment. Math. Helv. 82, 805–828 (2007)
Srivastava, A.M.: A course on Borel sets. Berlin-Heidleberg-New York: Springer, 1998
Sturm K.T.: On the geometry of metric measure spaces. I. Acta Math 196(1), 65–131 (2006)
Sturm K.T.: On the geometry of metric measure spaces. II. Acta Math 196(1), 133–177 (2006)
Sudakov V.N.: Geometric problems in the theory of dimensional distributions. Proc. Steklov Inst. Math. 141, 1–178 (1979)
Trudinger N., Wang X.J.: On the Monge mass transfer problem. Calc. Var. PDE 13, 19–31 (2001)
Villani, C.: Optimal transport, old and new. Berlin-Heidleberg-New York: Springer, 2008
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Communicated by P. Constantin
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Bianchini, S., Cavalletti, F. The Monge Problem for Distance Cost in Geodesic Spaces. Commun. Math. Phys. 318, 615–673 (2013). https://doi.org/10.1007/s00220-013-1663-8
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DOI: https://doi.org/10.1007/s00220-013-1663-8