Optimal transport from a point-like source

  • Franco CardinEmail author
  • Jayanth R. Banavar
  • Amos Maritan
Original Article


We present a dynamical interpretation of the Monge–Kantorovich theory in a stationary regime. This new principle, akin to the Fermat principle of geometric optics, captures the geodesic character of many distribution networks such as plant roots, river basins and the physiological transportation network of metabolites in living systems. Our general continuum framework allows us to map a previously proposed phenomenological principle into a stationary Monge optimization principle in the Kantorovich relaxed format.


Monge–Kantorovich Metabolic scaling Fermat principle 



We are indebted to Andrea Rinaldo for stimulating discussions and ongoing collaboration.


  1. 1.
    Banavar, J.R., Maritan, A., Rinaldo, A.: Size and form in efficient transportation networks. Nature 399, 130–132 (1999)ADSCrossRefGoogle Scholar
  2. 2.
    West, G.B., Woodruff, W.H., Brown, J.H.: Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals. Proc. Nat. Acad. Sci. 99, 2473–2478 (2002)ADSCrossRefGoogle Scholar
  3. 3.
    Kleiber, M.: Body size and metabolism. Hilgardia 6, 315–353 (1932)CrossRefGoogle Scholar
  4. 4.
    McMahon, T., Bonner, J.T.: On Size and Life, p. 255. Scientific American Books - W. H. Freeman & Co., New York (1983)Google Scholar
  5. 5.
    Dodds, P.S., Rothman, D.H., Weitz, J.S.: Re-examination of the 3/4-law of metabolism. J. Theor. Biol. 209, 9–27 (2001)CrossRefGoogle Scholar
  6. 6.
    Kolokotrones, T., et al.: Curvature in metabolic scaling. Nature 464, 753–756 (2010)ADSCrossRefGoogle Scholar
  7. 7.
    Dreyer, O., Puzio, R.: Allometric scaling in animals and plants. J. Math. Biol. 43, 144–156 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Tero, A., Kobayashi, R., Nakagaki, T.: A mathematical model for adaptive transport network in path finding by true slime mold. J. Theor. Biol. 244(4), 553 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Tero, A., Takagi, S., Saigusa, T., Ito, K., Bebber, D.P., Fricker, M.D., Yumiki, K., Kobayashi, R., Nakagaki, T.: Rules for biologically inspired adaptive network design. Science 327(5964), 439–442 (2010)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bonifaci, V., Mehlhorn, K., Varma, G.: Physarum can compute shortest paths. J. Theor. Biol. 309, 121–133 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Facca, E., Cardin, F., Putti, M.: Towards a stationary Monge–Kantorovich dynamics: the Physarum polycephalum experience. SIAM J. Appl. Math. 78(2), 651–676 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338, p. xxii+973. Springer, Berlin (2009)zbMATHGoogle Scholar
  13. 13.
    Santambrogio, F.: Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling. Progress in Nonlinear Differential Equations and their Applications, vol. 87. Birkhäuser, Cham (2015)zbMATHGoogle Scholar
  14. 14.
    Vershik, A.M.: Long history of the Monge–Kantorovich transportation problem. Math. Intell. 35(4), 1–9 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Monge, G.: Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pp. 666–704 (1781)Google Scholar
  16. 16.
    Brenier, Y.: Extended Monge–Kantorovich theory. In: Optimal Transportation and Applications. Lecture Notes in Mathematical, vol. 1813, pp. 91-121. Springer, Berlin (2003)CrossRefGoogle Scholar
  17. 17.
    Evans, L.C., Gangbo, W.: Differential equations methods for the Monge–Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137(653), viii+66 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Evans, L.C.: Partial Differential Equations and Monge–Kantorovich Mass Transfer, Current Developments in Mathematics. International Press, Boston (1999)Google Scholar
  19. 19.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)MathSciNetCrossRefGoogle Scholar
  20. 20.
    McCann, R.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ambrosio, L.: Lecture notes on optimal transport problems. In: Mathematical Aspects of Evolving Interfaces. Funchal 2000, Lecture Notes in Mathematics, vol. 1812. Springer, Berlin (2003)zbMATHGoogle Scholar
  23. 23.
    Buttazzo, G.: Evolution models for mass transportation problems. Milan J. Math. 80(1), 47–63 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ambrosio, L., Pratelli, A.: Existence and stability results in the \(L^1\) theory of optimal transportation. In: Caffarelli, L.A., Salsa, S. (eds.) LNM 1813, pp. 123–160 (2003)Google Scholar
  25. 25.
    Kantorovich, L.V.: On mass transportation. Dokl. Acad. Sci. USSR 37(7–8), 227–229 (1942). (in Russian)Google Scholar
  26. 26.
    Kantorovich, L.V.: Mathematical methods in the organization and planning of production. Reprint edition of the book, published in 1939, with introductory paper of L.V. Kantorovich. St. Petersburg, Publishing House of St. Petersburg University (2012)Google Scholar
  27. 27.
    Rinaldo, A., et al.: On feasible optimality. Istit. Veneto Sci. Lett. Arti Atti Cl. Sci. Fis. Mat. Natur. 155, 57–69 (1996–1997)Google Scholar
  28. 28.
    Facca, E., Cardin, F., Putti, M.: Physarum dynamics and optimal transport for basis pursuit. arXiv:1812.11782 (2018)

Copyright information

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Authors and Affiliations

  1. 1.Dipartimento di Matematica Tullio Levi-CivitaUniversità di PadovaPaduaItaly
  2. 2.Department of PhysicsUniversity of OregonEugeneUSA
  3. 3.Dipartimento di Fisica e Astronomia Galileo Galilei, INFNUniversità di PadovaPaduaItaly

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