A geometric study of Wasserstein spaces: Euclidean spaces
Kloeckner, Benoît1
1 Institut Fourier, 100 rue des Maths, BP 74, 38402 St Martin d’Hères, France
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 297-323.

In this article we consider Wasserstein spaces (with quadratic transportation cost) as intrinsic metric spaces. We are interested in usual geometric properties: curvature, rank and isometry group, mostly in the case of Euclidean spaces. Our most striking result is that the Wasserstein space of the line admits “exotic” isometries, which do not preserve the shape of measures.

Classification : 54E70, 28A33
Kloeckner, Benoît 1

1 Institut Fourier, 100 rue des Maths, BP 74, 38402 St Martin d’Hères, France 
@article{ASNSP_2010_5_9_2_297_0,
     author = {Kloeckner, Beno{\^\i}t},
     title = {A geometric study of {Wasserstein} spaces: {Euclidean} spaces},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {297--323},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {2},
     year = {2010},
     mrnumber = {2731158},
     zbl = {1218.53079},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2010_5_9_2_297_0/}
}
TY  - JOUR
AU  - Kloeckner, Benoît
TI  - A geometric study of Wasserstein spaces: Euclidean spaces
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2010
SP  - 297
EP  - 323
VL  - 9
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2010_5_9_2_297_0/
LA  - en
ID  - ASNSP_2010_5_9_2_297_0
ER  - 
%0 Journal Article
%A Kloeckner, Benoît
%T A geometric study of Wasserstein spaces: Euclidean spaces
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2010
%P 297-323
%V 9
%N 2
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2010_5_9_2_297_0/
%G en
%F ASNSP_2010_5_9_2_297_0
Kloeckner, Benoît. A geometric study of Wasserstein spaces: Euclidean spaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 297-323. http://www.numdam.org/item/ASNSP_2010_5_9_2_297_0/

[1] L. Ambrosio and N. Gigli, Construction of the parallel transport in the Wasserstein space, Methods Appl. Anal. 15 (2008), 1–30. | MR | Zbl

[2] L. Ambrosio, N. Gigli and G. Savaré, “Gradient Flows in Metric Spaces and in the Space of Probability Measures”, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. | MR | Zbl

[3] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 805–808. | MR | Zbl

[4] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375–417. | MR | Zbl

[5] M. R. Bridson and A. Haefliger, “Metric Spaces of Non-positive Curvature”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 319, Springer-Verlag, Berlin, 1999. | MR | Zbl

[6] D. Burago, Y. Burago and S. Ivanov, “A Course in Metric Geometry”, Graduate Studies in Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 2001. | MR | Zbl

[7] J.-M. Gambaudo and É. Ghys, Braids and signatures, Bull. Soc. Math. France 133 (2005), 541–579. | EuDML | Numdam | MR | Zbl

[8] S. Helgason, “The Radon Transform”, Progress in Mathematics, Vol. 5, Birkhäuser, Boston, Mass., 1980. | MR | Zbl

[9] J. Jost, “Nonpositive Curvature: Geometric and Analytic Aspects”, Lectures in Mathematics ETH Zürich, Birkhäuser, Verlag, Basel, 1997. | MR | Zbl

[10] N. Juillet, Optimal transport and geometric analysis in Heisenberg groups, Université Grenoble 1 et Rheinische Friedrich-Wilhelms-Universität Bonn, December 2008. | Zbl

[11] M. Knott and C. S. Smith, On the optimal mapping of distributions, J. Optim. Theory Appl. 43 (1984), 39–49. | MR | Zbl

[12] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), 903–991. | MR | Zbl

[13] J. Lott, Some geometric calculations on Wasserstein space, Comm. Math. Phys. 277 (2008), 423–437. | MR | Zbl

[14] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig 69 (1917), 262–277. English translation: Determination of a function from the values of its integrals along certain manifolds, In: “Computed Tomography” (Cincinnati, Ohio, 1982), Proc. Sympos. Appl. Math., Vol. 27, Amer. Math. Soc., Providence, R.I., 1982, 71–86. | JFM

[15] M.-K. Von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), 923–940. | MR | Zbl

[16] C. S. Smith and M. Knott, Note on the optimal transportation of distributions, J. Optim. Theory Appl. 52 (1987), 323–329. | MR | Zbl

[17] K.-T. Sturm, On the geometry of metric measure spaces, I, II, Acta Math. 196 (2006), 65–131, 133–177. | MR | Zbl

[18] A. Takatsu, On Wasserstein geometry of the space of Gaussian measures, arXiv: 0801.2250, 2008. | MR

[19] A. Takatsu and T. Yokota, Cone structure of L 2 -Wasserstein spaces, arXiv:0812.2752. | MR | Zbl

[20] C. Villani, “Topics in Optimal Transportation”, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. | MR | Zbl

[21] C. Villani, “Optimal Transport Old and New”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 338, Springer-Verlag, Berlin, 2009. | MR | Zbl