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We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target.
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Figalli, A. and Rifford, L. Continuity of optimal transport maps on small deformations of 𝕊 2 \mathbb{S}^{2} . Comm. Pure Appl. Math
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McCann, R.J. Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal
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Deneckere, R. and Severinov, S. Multi-dimensional screening with a one dimensional allocation space. Preprint
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Figalli, A., Kim, Y.-H. and McCann, R.J. Continuity and injectivity of optimal maps for non-negatively cross-curved costs. Preprint
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Figalli, A., Kim, Y.-H. and McCann, R.J. Regularity of optimal transport maps on multiple products of sphere. Preprint
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Figalli, A., Rifford, L. and Villani, C. Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manfifolds. Preprint
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Figalli, A., Rifford, L. and Villani, C. Nearly round spheres look convex. Preprint
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Kim, Y-H. and McCann, R.J. Continuity, curvature and the general covariance of optimal transportation. J. Eur. Math. Soc
2010
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