Snowflake universality  of Wasserstein spaces

Andoni, Alexandr; Naor, Assaf; Neiman, Ofer

doi:10.24033/asens.2363

Snowflake universality of Wasserstein spaces
[Universalité des espaces de Wasserstein à floconnage près]

Andoni, Alexandr ; Naor, Assaf ; Neiman, Ofer

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 657-700.

Résumé
Abstract

Pour $p \in (1, \infty)$ notons $𝒫_{p} (ℝ^{3})$ l'espace métrique des mesures de probabilité $p$ -intégrables sur $ℝ^{3}$ , muni de la $p$ -métrique de Wasserstein $𝖶_{p}$ . Nous montrons que pour tout $ε > 0$ , tout $θ \in (0, 1 / p]$ et tout espace métrique fini $(X, d_{X})$ , l'espace métrique $(X, d_{X}^{θ})$ se plonge dans $𝒫_{p} (ℝ^{3})$ avec distortion au plus $1 + ε$ . Nous montrons que cela est optimal quand $p \in (1, 2]$ au sens où l'exposant $1 / p$ ne peut pas être augmenté. En fait pour $n \in ℕ$ assez grand il existe un espace métrique à $n$ points $(X_{n}, d_{n})$ tel que pour tout $α \in (1 / p, 1]$ tout plongement de l'espace métrique $(X_{n}, d_{n}^{α})$ dans $𝒫_{p} (ℝ^{3})$ a une distortion au moins égale à un multiple par une constante de ${(log n)}^{α - 1 / p}$ . Ces résultats impliquent qu'il existe un espace d'Alexandrov de courbure positive, à savoir $𝒫_{2} (ℝ^{3})$ , vis- à-vis duquel il n'existe pas de suite de graphes expanseurs de degré borné. Il en résulte aussi que $𝒫_{2} (ℝ^{3})$ n'admet pas de plongement uniforme, grossier ou quasisymétrique dans un espace de Banach de type non trivial. Nous discutons le lien avec plusieurs questions ouvertes depuis longtemps en géométrie des espaces métriques, dont la caractérisation des sous-ensembles des espaces d'Alexandrov, l'existence d'expandeurs, le problème d'universalité pour $𝒫_{2} (ℝ^{k})$ , et le problème de dichotomie pour le cotype métrique.

For $p \in (1, \infty)$ let $𝒫_{p} (ℝ^{3})$ denote the metric space of all $p$ -integrable Borel probability measures on $ℝ^{3}$ , equipped with the Wasserstein $p$ metric $𝖶_{p}$ . We prove that for every $ε > 0$ , every $θ \in (0, 1 / p]$ and every finite metric space $(X, d_{X})$ , the metric space $(X, d_{X}^{θ})$ embeds into $𝒫_{p} (ℝ^{3})$ with distortion at most $1 + ε$ . We show that this is sharp when $p \in (1, 2]$ in the sense that the exponent $1 / p$ cannot be replaced by any larger number. In fact, for arbitrarily large $n \in ℕ$ there exists an $n$ -point metric space $(X_{n}, d_{n})$ such that for every $α \in (1 / p, 1]$ any embedding of the metric space $(X_{n}, d_{n}^{α})$ into $𝒫_{p} (ℝ^{3})$ incurs distortion that is at least a constant multiple of ${(log n)}^{α - 1 / p}$ . These statements establish that there exists an Alexandrov space of nonnegative curvature, namely $𝒫_{2} (ℝ^{3})$ , with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that $𝒫_{2} (ℝ^{3})$ does not admit a uniform, coarse, or quasisymmetric embedding into any Banach space of nontrivial type. Links to several longstanding open questions in metric geometry are discussed, including the characterization of subsets of Alexandrov spaces, existence of expanders, the universality problem for $𝒫_{2} (ℝ^{k})$ , and the metric cotype dichotomy problem.

Publié le : 2018-11-12

DOI : 10.24033/asens.2363

Classification : 46B85, 53C23, 46E27.
Keywords: Metric embeddings, Wasserstein spaces, Alexandrov spaces, Snowflakes of metric spaces, nonlinear spectral gaps, metric cotype, Markov type.
Mot clés : Plongements d'espaces métriques, espaces de Wasserstein, espaces d'Alexandrov, floconnage d'espaces métriques, trou spectral non linéaire, cotype métrique, type de Markov.

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     author = {Andoni, Alexandr and Naor, Assaf and Neiman, Ofer},
     title = {Snowflake universality  of {Wasserstein} spaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {657--700},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {3},
     year = {2018},
     doi = {10.24033/asens.2363},
     mrnumber = {3831034},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2363/}
}

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Andoni, Alexandr; Naor, Assaf; Neiman, Ofer. Snowflake universality  of Wasserstein spaces. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 657-700. doi : 10.24033/asens.2363. https://www.numdam.org/articles/10.24033/asens.2363/

Bibliographie
Cité par

Alexander, S.; Kapovitch, V.; Petrunin, A. Alexandrov meets Kirszbraun, Proceedings of the Gökova Geometry-Topology Conference 2010, Int. Press, Somerville, MA (2011), pp. 88-109 | MR | Zbl

Austin, T.; Naor, A. On the bi-Lipschitz structure of Wasserstein spaces (2015) (preprint)

Andoni, A.; Naor, A.; Neiman, O. Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) (Chatzigiannakis, I.; Mitzenmacher, M.; Sangiorgi, D., eds.) (Leibniz International Proceedings in Informatics (LIPIcs)), Volume 55 (2016), p. 83:1-83:14 | MR

Austin, T.; Naor, A.; Peres, Y. The wreath product of $ℤ$ with $ℤ$ has Hilbert compression exponent $\frac{2}{3}$ , Proc. Amer. Math. Soc., Volume 137 (2009), pp. 85-90 (ISSN: 0002-9939) | DOI | MR | Zbl

Beurling, A.; Ahlfors, L. The boundary correspondence under quasiconformal mappings, Acta Math., Volume 96 (1956), pp. 125-142 (ISSN: 0001-5962) | DOI | MR | Zbl

Ball, K. Markov chains, Riesz transforms and Lipschitz maps, Geom. Funct. Anal., Volume 2 (1992), pp. 137-172 (ISSN: 1016-443X) | DOI | MR | Zbl

Bartal, Y., 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, pp. 184-193 | DOI | MR

Buckley, S. M.; Falk, K.; Wraith, D. J. Ptolemaic spaces and CAT(0), Glasg. Math. J., Volume 51 (2009), pp. 301-314 (ISSN: 0017-0895) | DOI | MR | Zbl

Burago, Y.; Gromov, M.; Perel'man, G. A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk, Volume 47 (1992), p. 3-51, 222 (ISSN: 0042-1316) | DOI | MR | Zbl

Bridson, M. R.; Haefliger, A., Grundl. math. Wiss., 319, Springer, Berlin, 1999, 643 pages (ISBN: 3-540-64324-9) | DOI | MR | Zbl

Bartal, Y.; Linial, N.; Mendel, M.; Naor, A. On metric Ramsey-type phenomena, Ann. of Math., Volume 162 (2005), pp. 643-709 (ISSN: 0003-486X) | DOI | MR | Zbl

Bourgain, J.; Milman, V. Dichotomie du cotype pour les espaces invariants, C. R. Acad. Sci. Paris Sér. I Math., Volume 300 (1985), pp. 263-266 (ISSN: 0249-6291) | MR | Zbl

Berg, I. D.; Nikolaev, I. G. On a distance characterization of A. D. Aleksandrov spaces of nonpositive curvature, Dokl. Akad. Nauk, Volume 414 (2007), pp. 10-12 (ISSN: 0869-5652) | DOI | MR | Zbl

Berg, I. D.; Nikolaev, I. G. Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, Volume 133 (2008), pp. 195-218 (ISSN: 0046-5755) | DOI | MR | Zbl

Briët, J.; Naor, A.; Regev, O. Locally decodable codes and the failure of cotype for projective tensor products, Electron. Res. Announc. Math. Sci., Volume 19 (2012), pp. 120-130 (ISSN: 1935-9179) | DOI | MR | Zbl

Bourgain, J. New Banach space properties of the disc algebra and $H^{\infty}$ , Acta Math., Volume 152 (1984), pp. 1-48 (ISSN: 0001-5962) | DOI | MR | Zbl

Bourgain, J. On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math., Volume 52 (1985), pp. 46-52 (ISSN: 0021-2172) | DOI | MR | Zbl

Bourgain, J. The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math., Volume 56 (1986), pp. 222-230 (ISSN: 0021-2172) | DOI | MR | Zbl

Busemann, H., Academic Press Inc., New York, N. Y., 1955, 422 pages | MR | Zbl

Critchley, F.; Fichet, B., Classification and dissimilarity analysis (Lect. Notes Stat.), Volume 93, Springer, New York, 1994, pp. 5-65 | DOI | MR | Zbl

Ding, J.; Lee, J. R.; Peres, Y. Markov type and threshold embeddings, Geom. Funct. Anal., Volume 23 (2013), pp. 1207-1229 | DOI | MR | Zbl

David, G.; Semmes, S., Oxford Lecture Series in Mathematics and its Applications, 7, The Clarendon Press, Oxford Univ. Press, New York, 1997, 212 pages (ISBN: 0-19-850166-8) | MR | Zbl

Enflo, P. On the nonexistence of uniform homeomorphisms between $L_{p}$ -spaces, Ark. Mat., Volume 8 (1969), pp. 103-105 (ISSN: 0004-2080) | DOI | MR | Zbl

Enflo, P. Uniform homeomorphisms between Banach spaces, Séminaire Maurey-Schwartz (1975–1976), Espaces, $L^{p}$ , applications radonifiantes et géométrie des espaces de Banach, Exp. No. 18, Centre Math., École polytech., Palaiseau (1976) | Numdam | MR | Zbl

Foertsch, T.; Lytchak, A.; Schroeder, V. Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not., Volume 2007 (2007) (ISSN: 1073-7928) | DOI | MR | Zbl

Foertsch, T.; Schroeder, V. Hyperbolicity, $CAT (- 1)$ -spaces and the Ptolemy inequality, Math. Ann., Volume 350 (2011), pp. 339-356 (ISSN: 0025-5831) | DOI | MR | Zbl

Garling, D. J. H., Cambridge Univ. Press, Cambridge, 2007, 335 pages (ISBN: 978-0-521-69973-0) | DOI | MR | Zbl

Godefroy, G.; Kalton, N. J. Lipschitz-free Banach spaces, Studia Math., Volume 159 (2003), pp. 121-141 (ISSN: 0039-3223) | DOI | MR | Zbl

Gromov, M. $CAT (κ)$ -spaces: construction and concentration, Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 280 (2001) (ISSN: 0373-2703) | DOI | MR | Zbl

Gromov, M. Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003), pp. 73-146 (ISSN: 1016-443X) | DOI | MR | Zbl

Gromov, M. Filling Riemannian manifolds, J. Differential Geom., Volume 18 (1983), pp. 1-147 http://projecteuclid.org/euclid.jdg/1214509283 (ISSN: 0022-040X) | MR | Zbl

Gromov, M., Geometric group theory, vol. 2 (Sussex, 1991) (London Math. Soc. Lecture Note Ser.), Volume 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295 | MR | Zbl

Gromov, M., Progress in Math., 152, Birkhäuser, 1999, 585 pages (ISBN: 0-8176-3898-9) | MR | Zbl

Hoory, S.; Linial, N.; Wigderson, A. Expander graphs and their applications, Bull. Amer. Math. Soc., Volume 43 (2006), pp. 439-561 (ISSN: 0273-0979) | DOI | MR | Zbl

Har-Peled, S.; Mendel, M. Fast construction of nets in low-dimensional metrics and their applications, SIAM J. Comput., Volume 35 (2006), pp. 1148-1184 (ISSN: 0097-5397) | DOI | MR | Zbl

Jost, J., Lectures in Mathematics ETH Zürich, Birkhäuser, 1997, 108 pages (ISBN: 3-7643-5736-3) | DOI | MR | Zbl

Kay, D. C. The ptolemaic inequality in Hilbert geometries, Pacific J. Math., Volume 21 (1967), pp. 293-301 http://projecteuclid.org/euclid.pjm/1102992501 (ISSN: 0030-8730) | DOI | MR | Zbl

Kleiner, B.; Leeb, B. Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Publ. Math. IHÉS, Volume 86 (1997), pp. 115-197 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Khot, S.; Naor, A. Nonembeddability theorems via Fourier analysis, Math. Ann., Volume 334 (2006), pp. 821-852 (ISSN: 0025-5831) | DOI | MR | Zbl

Kwapień, S.; Pełczyński, A. Absolutely summing operators and translation-invariant spaces of functions on compact abelian groups, Math. Nachr., Volume 94 (1980), pp. 303-340 (ISSN: 0025-584X) | DOI | MR | Zbl

Laakso, T. J. Plane with $A_{\infty}$ -weighted metric not bi-Lipschitz embeddable to $ℝ^{N}$ , Bull. London Math. Soc., Volume 34 (2002), pp. 667-676 (ISSN: 0024-6093) | DOI | MR | Zbl

Lafforgue, V. Un renforcement de la propriété (T), Duke Math. J., Volume 143 (2008), pp. 559-602 (ISSN: 0012-7094) | DOI | MR | Zbl

Lafforgue, V. Propriété (T) renforcée banachique et transformation de Fourier rapide, J. Topol. Anal., Volume 1 (2009), pp. 191-206 (ISSN: 1793-5253) | DOI | MR | Zbl

Liao, B. Strong Banach property (T) for simple algebraic groups of higher rank, J. Topol. Anal., Volume 6 (2014), pp. 75-105 (ISSN: 1793-5253) | DOI | MR | Zbl

Linial, N.; Magen, A.; Naor, A. Girth and Euclidean distortion, Geom. Funct. Anal., Volume 12 (2002), pp. 380-394 (ISSN: 1016-443X) | DOI | MR | Zbl

Lee, J. R.; Mendel, M.; Naor, A. Metric structures in $L_{1}$ : dimension, snowflakes, and average distortion, European J. Combin., Volume 26 (2005), pp. 1180-1190 (ISSN: 0195-6698) | DOI | MR | Zbl

Lee, J. R.; Naor, A. Embedding the diamond graph in $L_{p}$ and dimension reduction in $L_{1}$ , Geom. Funct. Anal., Volume 14 (2004), pp. 745-747 (ISSN: 1016-443X) | DOI | MR | Zbl

Lang, U.; Plaut, C. Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata, Volume 87 (2001), pp. 285-307 (ISSN: 0046-5755) | DOI | MR | Zbl

Lebedeva, N.; Petrunin, A. Curvature bounded below: a definition a la Berg-Nikolaev, Electron. Res. Announc. Math. Sci., Volume 17 (2010), pp. 122-124 (ISSN: 1935-9179) | DOI | MR | Zbl

Lang, U.; Pavlović, B.; Schroeder, V. Extensions of Lipschitz maps into Hadamard spaces, Geom. Funct. Anal., Volume 10 (2000), pp. 1527-1553 (ISSN: 1016-443X) | DOI | MR | Zbl

Lang, U.; Schroeder, V. Kirszbraun's theorem and metric spaces of bounded curvature, Geom. Funct. Anal., Volume 7 (1997), pp. 535-560 (ISSN: 1016-443X) | DOI | MR | Zbl

Lott, J.; Villani, C. Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., Volume 169 (2009), pp. 903-991 (ISSN: 0003-486X) | DOI | MR | Zbl

Matoušek, J., Graduate Texts in Math., 212, Springer, 2002 | MR | Zbl

Matoušek, J. On embedding expanders into $l_{p}$ spaces, Israel J. Math., Volume 102 (1997), pp. 189-197 (ISSN: 0021-2172) | DOI | MR | Zbl

Maurey, B. Type, cotype and $K$ -convexity, Handbook of the geometry of Banach spaces, Volume 2, North-Holland (2003), pp. 1299-1332 | DOI | MR | Zbl

Mendel, M., Limits of graphs in group theory and computer science, EPFL Press, Lausanne, 2009, pp. 59-76 | MR | Zbl

Mimura, M. Sphere equivalence, Banach expanders, and extrapolation, Int. Math. Res. Not., Volume 2015 (2015), pp. 4372-4391 (ISSN: 1073-7928) | DOI | MR

Mendel, M.; Naor, A. A note on dichotomies for metric transforms (preprint arXiv:1102.1800 )

Mendel, M.; Naor, A. Metric cotype, Ann. of Math., Volume 168 (2008), pp. 247-298 (ISSN: 0003-486X) | DOI | MR | Zbl

Mendel, M.; Naor, A. Maximum gradient embeddings and monotone clustering, Combinatorica, Volume 30 (2010), pp. 581-615 (ISSN: 0209-9683) | DOI | MR | Zbl

Mendel, M.; Naor, A. Markov convexity and local rigidity of distorted metrics, J. Eur. Math. Soc. (JEMS), Volume 15 (2013), pp. 287-337 (ISSN: 1435-9855) | DOI | MR | Zbl

Mendel, M.; Naor, A. Nonlinear spectral calculus and super-expanders, Publ. Math. IHÉS, Volume 119 (2014), pp. 1-95 (ISSN: 0073-8301) | DOI | MR | Zbl

Mendel, M.; Naor, A. Expanders with respect to Hadamard spaces and random graphs, Duke Math. J., Volume 164 (2015), pp. 1471-1548 (ISSN: 0012-7094) | DOI | MR

Maurey, B.; Pisier, G. Caractérisation d'une classe d'espaces de Banach par des propriétés de séries aléatoires vectorielles, C. R. Acad. Sci. Paris, Volume 277 (1973), p. A687-A690 | MR | Zbl

Naor, A. A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between $L_{p}$ spaces, Mathematika, Volume 48 (2001), pp. 253-271 (ISSN: 0025-5793) | DOI | MR | Zbl

Naor, A., Metric and differential geometry (Progr. Math.), Volume 297, Birkhäuser, 2012, pp. 175-178 | DOI | MR | Zbl

Naor, A. An introduction to the Ribe program, Jpn. J. Math., Volume 7 (2012), pp. 167-233 (ISSN: 0289-2316) | DOI | MR | Zbl

Naor, A. Comparison of metric spectral gaps, Anal. Geom. Metr. Spaces, Volume 2 (2014), pp. 1-52 (ISSN: 2299-3274) | DOI | MR | Zbl

Nowak, P. W. Poincaré inequalities and rigidity for actions on Banach spaces, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 689-709 (ISSN: 1435-9855) | DOI | MR

Naor, A.; Peres, Y. Embeddings of discrete groups and the speed of random walks, Int. Math. Res. Not., Volume 2008 (2008) (ISSN: 1073-7928) | DOI | MR | Zbl

Naor, A.; Peres, Y. $L_{p}$ compression, traveling salesmen, and stable walks, Duke Math. J., Volume 157 (2011), pp. 53-108 (ISSN: 0012-7094) | DOI | MR | Zbl

Naor, A.; Peres, Y.; Schramm, O.; Sheffield, S. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J., Volume 134 (2006), pp. 165-197 (ISSN: 0012-7094) | DOI | MR | Zbl

Newman, I.; Rabinovich, Y. A lower bound on the distortion of embedding planar metrics into Euclidean space, Discrete Comput. Geom., Volume 29 (2003), pp. 77-81 (ISSN: 0179-5376) | DOI | MR | Zbl

Naor, A.; Schechtman, G. Remarks on non linear type and Pisier's inequality, J. reine angew. Math., Volume 552 (2002), pp. 213-236 (ISSN: 0075-4102) | DOI | MR | Zbl

Naor, A.; Schechtman, G. Planar earthmover is not in $L_{1}$ , SIAM J. Comput., Volume 37 (2007), pp. 804-826 (ISSN: 0097-5397) | DOI | MR | Zbl

Naor, A.; Silberman, L. Poincaré inequalities, embeddings, and wild groups, Compos. Math., Volume 147 (2011), pp. 1546-1572 (ISSN: 0010-437X) | DOI | MR | Zbl

Ohta, S.-i. Markov type of Alexandrov spaces of non-negative curvature, Mathematika, Volume 55 (2009), pp. 177-189 (ISSN: 0025-5793) | DOI | MR | Zbl

Ohta, S.-i.; Pichot, M. A note on Markov type constants, Arch. Math. (Basel), Volume 92 (2009), pp. 80-88 (ISSN: 0003-889X) | DOI | MR | Zbl

Otto, F. The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, Volume 26 (2001), pp. 101-174 (ISSN: 0360-5302) | DOI | MR | Zbl

Ozawa, N. A note on non-amenability of $ℬ (l_{p})$ for $p = 1, 2$ , Internat. J. Math., Volume 15 (2004), pp. 557-565 (ISSN: 0129-167X) | DOI | MR | Zbl

Pełczyński, A., Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, 30, Amer. Math. Soc., 1977 | MR | Zbl

Pisier, G., Séminaire sur la Géométrie des Espaces de Banach (1977–1978), exp. n^o 14, École polytech., Palaiseau, 1978 | Numdam | MR | Zbl

Pisier, G. Factorization of operator valued analytic functions, Adv. Math., Volume 93 (1992), pp. 61-125 (ISSN: 0001-8708) | DOI | MR | Zbl

Rešetnjak, J. G. Non-expansive maps in a space of curvature no greater than $K$ , Sibirsk. Mat. Ž., Volume 9 (1968), pp. 918-927 (ISSN: 0037-4474) | MR

Ribe, M. On uniformly homeomorphic normed spaces, Ark. Mat., Volume 14 (1976), pp. 237-244 (ISSN: 0004-2080) | DOI | MR | Zbl

Rachev, S. T.; Rüschendorf, L., Probability and its Applications (New York), Springer, New York, 1998, 430 pages (ISBN: 0-387-98352-X) | MR | Zbl

Sato, T. An alternative proof of Berg and Nikolaev's characterization of $CAT (0)$ -spaces via quadrilateral inequality, Arch. Math. (Basel), Volume 93 (2009), pp. 487-490 (ISSN: 0003-889X) | DOI | MR | Zbl

Schoenberg, I. Metric spaces and positive definite functions, Trans. Amer. Math. Soc., Volume 44 (1938), pp. 522-536 | DOI | JFM | MR

Sturm, K.-T., Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) (Contemp. Math.), Volume 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357-390 | DOI | MR | Zbl

Sturm, K.-T. On the geometry of metric measure spaces. I, Acta Math., Volume 196 (2006), pp. 65-131 (ISSN: 0001-5962) | DOI | MR | Zbl

Sturm, K.-T. Metric spaces of lower bounded curvature, Exposition. Math., Volume 17 (1999), pp. 35-47 (ISSN: 0723-0869) | MR | Zbl

Thurston, D. Length inequalities in trees and CAT(0) spaces (2014) ( http://mathoverflow.net/q/163706 )

Tomczak-Jaegermann, N. The moduli of smoothness and convexity and the Rademacher averages of trace classes $S_{p} (1 \leq p < \infty)$ , Studia Math., Volume 50 (1974), pp. 163-182 (ISSN: 0039-3223) | MR | Zbl

Tukia, P.; Väisälä, J. Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 5 (1980), pp. 97-114 (ISSN: 0066-1953) | DOI | MR | Zbl

Varopoulos, N. Une remarque sur les ensembles de Helson, Duke Math. J., Volume 43 (1976), pp. 387-390 http://projecteuclid.org/euclid.dmj/1077311648 (ISSN: 0012-7094) | DOI | MR | Zbl

Villani, C., Graduate Studies in Math., 58, Amer. Math. Soc., Providence, RI, 2003, 370 pages (ISBN: 0-8218-3312-X) | DOI | MR | Zbl

Wojtaszczyk, P., Cambridge Studies in Advanced Math., 25, Cambridge Univ. Press, Cambridge, 1991, 382 pages (ISBN: 0-521-35618-0) | DOI | MR | Zbl

Wells, J. H.; Williams, L. R., Ergebn. Math. Grenzg., 84, Springer, 1975, 108 pages | MR | Zbl

Yokota, T. A rigidity theorem in Alexandrov spaces with lower curvature bound, Math. Ann., Volume 353 (2012), pp. 305-331 (ISSN: 0025-5831) | DOI | MR | Zbl

Cité par Sources :