Book Review
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MathSciNet review:
2721045
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Book Information:
Author:
Cédric Villani
Title:
Optimal transport: old and new
Additional book information:
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338,
Springer-Verlag, Berlin,
2009,
xxii+973 pp.,
ISBN 978-3-540-71049-3,
US $159.00
Yann Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 19, 805–808 (French, with English summary). MR 923203
Yann Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375–417. MR 1100809, DOI 10.1002/cpa.3160440402
Luis A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc. 5 (1992), no. 1, 99–104. MR 1124980, DOI 10.1090/S0894-0347-1992-1124980-8
A. Figalli. Regularity of optimal transport maps (after Ma-Trudinger-Wang and Loeper). Séminaire Bourbaki. Vol. 2008/2009. Exp. No. 1009.
L. V. Kantorovich. On mass transportation. Dokl. Akad. Nauk. SSSR, 37 (1942), 227-229.
L. V. Kantorovich. On a problem of Monge. Uspekhi Mat. Nauk., 3 (1948), 225-226.
Grégoire Loeper, On the regularity of solutions of optimal transportation problems, Acta Math. 202 (2009), no. 2, 241–283. MR 2506751, DOI 10.1007/s11511-009-0037-8
John Lott and Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. MR 2480619, DOI 10.4007/annals.2009.169.903
Xi-Nan Ma, Neil S. Trudinger, and Xu-Jia Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal. 177 (2005), no. 2, 151–183. MR 2188047, DOI 10.1007/s00205-005-0362-9
Robert J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), no. 1, 153–179. MR 1451422, DOI 10.1006/aima.1997.1634
Robert J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11 (2001), no. 3, 589–608. MR 1844080, DOI 10.1007/PL00001679
G. Monge. Mémoire sur la Théorie des Déblais et des Remblais. Hist. de l’Acad. des Sciences de Paris (1781), 666-704.
Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), no. 1-2, 101–174. MR 1842429, DOI 10.1081/PDE-100002243
Karl-Theodor Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65–131. MR 2237206, DOI 10.1007/s11511-006-0002-8
Karl-Theodor Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 133–177. MR 2237207, DOI 10.1007/s11511-006-0003-7
Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483, DOI 10.1090/gsm/058
References
- Y. Brenier. Polar decomposition and increasing rearrangement of vector fields. C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), no. 19, 805-808. MR 923203 (89b:58226)
- Y. Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math., 44 (1991), 375–417. MR 1100809 (92d:46088)
- L. A. Caffarelli. The regularity of mappings with a convex potential. J. Amer. Math. Soc., 5 (1992), no. 1, 99-104. MR 1124980 (92j:35018)
- A. Figalli. Regularity of optimal transport maps (after Ma-Trudinger-Wang and Loeper). Séminaire Bourbaki. Vol. 2008/2009. Exp. No. 1009.
- L. V. Kantorovich. On mass transportation. Dokl. Akad. Nauk. SSSR, 37 (1942), 227-229.
- L. V. Kantorovich. On a problem of Monge. Uspekhi Mat. Nauk., 3 (1948), 225-226.
- G. Loeper. On the regularity of solutions of optimal transportation problems. Acta Math., to appear. MR 2506751
- J. Lott and C. Villani. Ricci curvature via optimal transport. Ann. of Math. (2), 169 (2009), no. 3, 903-991. MR 2480619
- X. N. Ma, N. S. Trudinger and X. J. Wang. Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal., 177 (2005), no. 2, 151-183. MR 2188047 (2006m:35105)
- R. J. McCann. A convexity principle for interacting gases. Adv. Math., 128 (1997), no. 1, 153-179. MR 1451422 (98e:82003)
- R. J. McCann. Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal., 11 (2001), no. 3, 589-608. MR 1844080 (2002g:58017)
- G. Monge. Mémoire sur la Théorie des Déblais et des Remblais. Hist. de l’Acad. des Sciences de Paris (1781), 666-704.
- F. Otto. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations, 26 (2001), 101-174. MR 1842429 (2002j:35180)
- K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math., 196 (2006), no. 1, 65-131. MR 2237206 (2007k:53051a)
- K.-T. Sturm. On the geometry of metric measure spaces. II. Acta Math., 196 (2006), no. 1, 133-177. MR 2237207 (2007k:53051b)
- C. Villani. Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. MR 1964483 (2004e:90003)
Review Information:
Reviewer:
Alessio Figalli
Affiliation:
The University of Texas at Austin
Email:
figalli@math.utexas.edu
Journal:
Bull. Amer. Math. Soc.
47 (2010), 723-727
DOI:
https://doi.org/10.1090/S0273-0979-10-01285-1
Published electronically:
February 9, 2010
Review copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
