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Partial Regularity Results in Optimal Transportation

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Trends in Contemporary Mathematics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 8))

Abstract

This note describes some recent results on the regularity of optimal transport maps. As we shall see, in general optimal maps are not globally smooth, but they are so outside a “singular set” of measure zero.

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Notes

  1. 1.

    Here and in the sequel, | E | denotes the Lebesgue measure of a set E.

  2. 2.

    By an example of Pogorelov this turns out to be a necessary condition, see for instance [24, Section 4.1.3].

  3. 3.

    Actually, in [12, 14] the regular set is defined in a slightly different way and it is in principle smaller. However, the advantage of that definition is that it allows for a more refined analysis of the singular set (see [12]).

  4. 4.

    Although we did not state them here, many existence and uniqueness result for optimal transport maps on Riemannian manifolds are known (see for instance [11]), and they include for instance the case \(c(x,y) = d(x,y)^{2}/2\).

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Authors and Affiliations

  1. Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    G. De Philippis

  2. Department of Mathematics, The University of Texas at Austin, 2515 Speedway Stop C1200, Austin, TX, 78712-1202, USA

    A. Figalli

Authors
  1. G. De Philippis
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  2. A. Figalli
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Correspondence to G. De Philippis .

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Editors and Affiliations

  1. Istituto Nazionale di Alta Matematica "Francesco Severi", Roma, Italy

    Vincenzo Ancona

  2. Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Roma, Italy

    Elisabetta Strickland

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De Philippis, G., Figalli, A. (2014). Partial Regularity Results in Optimal Transportation. In: Ancona, V., Strickland, E. (eds) Trends in Contemporary Mathematics. Springer INdAM Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-05254-0_21

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