Interior regularity of optimal transport paths

  • Qinglan Xia


In a previous paper [12], we considered problems for which the cost of transporting one probability measure to another is given by a transport path rather than a transport map. In this model overlapping transport is frequently more economical. In the present article we study the interior regularity properties of such optimal transport paths. We prove that an optimal transport path of finite cost is rectifiable and simply a finite union of line segments near each interior point of the path.


Probability Measure Line Segment Present Article Interior Point Regularity Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allard, W.K., Almgren, F.J., Jr.: The structure of stationary one dimensional varifolds with positive density. Invent. Math. 34, 83-97 (1976)zbMATHGoogle Scholar
  2. 2.
    Ambrosio, L.: Lecture notes on optimal transport problems. Scuola Normale Superiore, Pisa (Preprint) 2000Google Scholar
  3. 3.
    Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137, no. 653 (1999)Google Scholar
  4. 4.
    Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York 1969Google Scholar
  5. 5.
    Fleming, W.: Flat chains over a finite coefficient group. Trans. Amer. Math. Soc. 121, 160-186 (1966)zbMATHGoogle Scholar
  6. 6.
    Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Monge, G.: Mémoire sur la théorie des déblais et de remblais, Histoire de l’Académie Royale des Sciences de Paris, avec les Mémorires de Mathématique et de Physique pour la même ann ée, pp. 666-704 (1781)Google Scholar
  8. 8.
    Kantorovich, L.: On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.) 37, 199-201 (1942)Google Scholar
  9. 9.
    Simon, L.: Lectures on geometric measure theory. Proc. Centre Math. Anal. Australian National University. 3 (1983)Google Scholar
  10. 10.
    Sudakov, V.N.: Geometric problems in the theory of infinite-dimensional probability distributions. Cover to cover translation of Trudy Mat. Inst. Steklov 141 (1976). Proc. Steklov Inst. Math. 2 (i-v), 1-178 (1979)Google Scholar
  11. 11.
    White, B.: Rectifiability of flat chains. Ann. of Math. (2) 150, 165-184 (1999)Google Scholar
  12. 12.
    Xia, Q.: Optimal paths related to transport problems. Communications in Contemporary Mathematics 5, 251-279 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  • Qinglan Xia
    • 1
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA

Personalised recommendations