Book Review
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MathSciNet review: 3497795
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Book Information:
Author: M. I. Ostrovskii
Title: Metric embeddings: bilipschitz and coarse embedddings into Banach spaces
Additional book information: de Gruyter Studies in Mathematics, Vol. 49, de Gruyter, Berlin, 2013, xii+372 pp., ISBN 978-3-11-026401-2, US$154.00.
- Israel Aharoni, Every separable metric space is Lipschitz equivalent to a subset of $c^{+}_{0}$, Israel J. Math. 19 (1974), 284–291. MR 511661, DOI https://doi.org/10.1007/BF02757727
- Sanjeev Arora, James R. Lee, and Assaf Naor, Euclidean distortion and the sparsest cut, J. Amer. Math. Soc. 21 (2008), no. 1, 1–21. MR 2350049, DOI https://doi.org/10.1090/S0894-0347-07-00573-5
- Patrice Assouad, Remarques sur un article de Israel Aharoni sur les prolongements lipschitziens dans $c_{0}$ (Israel J. Math. 19 (1974), 284–291), Israel J. Math. 31 (1978), no. 1, 97–100. MR 511662, DOI https://doi.org/10.1007/BF02761384
- Tim Austin, Assaf Naor, and Yuval Peres, The wreath product of $\Bbb Z$ with $\Bbb Z$ has Hilbert compression exponent $\frac {2}{3}$, Proc. Amer. Math. Soc. 137 (2009), no. 1, 85–90. MR 2439428, DOI https://doi.org/10.1090/S0002-9939-08-09501-4
- K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geom. Funct. Anal. 2 (1992), no. 2, 137–172. MR 1159828, DOI https://doi.org/10.1007/BF01896971
- Keith Ball, The Ribe programme, Astérisque 352 (2013), Exp. No. 1047, viii, 147–159. Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058. MR 3087345
- Stefan Banach, Théorie des opérations linéaires, Éditions Jacques Gabay, Sceaux, 1993 (French). Reprint of the 1932 original. MR 1357166
- Florent Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Arch. Math. (Basel) 89 (2007), no. 5, 419–429. MR 2363693, DOI https://doi.org/10.1007/s00013-007-2108-4
- F. Baudier and G. Lancien, Tight embeddability of proper and stable metric spaces, Anal. Geom. Metr. Spaces 3 (2015), no. 1, 140–156. MR 3365754, DOI https://doi.org/10.1515/agms-2015-0010
- Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673
- C. Bessaga and A. Pełczyński, On the topological classification of complete linear metric spaces., General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 87–90. MR 0145324
- J. Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math. 52 (1985), no. 1-2, 46–52. MR 815600, DOI https://doi.org/10.1007/BF02776078
- J. Bourgain, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math. 56 (1986), no. 2, 222–230. MR 880292, DOI https://doi.org/10.1007/BF02766125
- J. Bourgain, V. Milman, and H. Wolfson, On type of metric spaces, Trans. Amer. Math. Soc. 294 (1986), no. 1, 295–317. MR 819949, DOI https://doi.org/10.1090/S0002-9947-1986-0819949-8
- Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
- Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
- Per Enflo, On the nonexistence of uniform homeomorphisms between $L_{p}$-spaces, Ark. Mat. 8 (1969), 103–105. MR 271719, DOI https://doi.org/10.1007/BF02589549
- M. Fréchet, Sur quelques points du calcul fonctionel, Rend. Circ. Mat. Palermo Math. 22 (1906), 1–71.
- M. Fréchet, Les dimensions d’un ensemble abstrait, Mathematische Annalen 68 (1910), no. 2, 145–168.
- Maurice Fréchet, Les espaces abstraits, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1989 (French). Reprint of the 1928 original. MR 1189135
- G. Godefroy, G. Lancien, and V. Zizler, The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain J. Math. 44 (2014), no. 5, 1529–1583. MR 3295641, DOI https://doi.org/10.1216/RMJ-2014-44-5-1529
- Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
- M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
- M. Gromov, in: S. Ferry, A. Ranicki, J. Rosenberg (Eds.), Problems (4) and (5),, Novikov Conjectures, Index Theorems and Rigidity, Vol. 1 (Oberwolfach, 1993), 1995, p. 67.
- S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), no. 3, 225–251. MR 675426, DOI https://doi.org/10.4064/sm-73-3-225-251
- William B. Johnson and Joram Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 189–206. MR 737400, DOI https://doi.org/10.1090/conm/026/737400
- William B. Johnson and N. Lovasoa Randrianarivony, $l_p\ (p>2)$ does not coarsely embed into a Hilbert space, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1045–1050. MR 2196037, DOI https://doi.org/10.1090/S0002-9939-05-08415-7
- William B. Johnson and Gideon Schechtman, Diamond graphs and super-reflexivity, J. Topol. Anal. 1 (2009), no. 2, 177–189. MR 2541760, DOI https://doi.org/10.1142/S1793525309000114
- M. I. Kadets, A proof of the topological equivalence of all separable infinite-dimensional Banach spaces, Funkcional. Anal. i Priložen. 1 (1967), 61–70 (Russian).
- N. J. Kalton, Coarse and uniform embeddings into reflexive spaces, Q. J. Math. 58 (2007), no. 3, 393–414. MR 2354924, DOI https://doi.org/10.1093/qmath/ham018
- N. J. Kalton, Lipschitz and uniform embeddings into $\ell _\infty $, Fund. Math. 212 (2011), no. 1, 53–69. MR 2771588, DOI https://doi.org/10.4064/fm212-1-4
- N. J. Kalton, The uniform structure of Banach spaces, Math. Ann. 354 (2012), no. 4, 1247–1288. MR 2992997, DOI https://doi.org/10.1007/s00208-011-0743-3
- N. J. Kalton, Uniform homeomorphisms of Banach spaces and asymptotic structure, Trans. Amer. Math. Soc. 365 (2013), no. 2, 1051–1079. MR 2995383, DOI https://doi.org/10.1090/S0002-9947-2012-05665-0
- N. J. Kalton, Examples of uniformly homeomorphic Banach spaces, Israel J. Math. 194 (2013), no. 1, 151–182. MR 3047066, DOI https://doi.org/10.1007/s11856-012-0080-6
- N. J. Kalton and G. Lancien, Best constants for Lipschitz embeddings of metric spaces into $c_0$, Fund. Math. 199 (2008), no. 3, 249–272. MR 2395263, DOI https://doi.org/10.4064/fm199-3-4
- Gennadi Kasparov and Guoliang Yu, The coarse geometric Novikov conjecture and uniform convexity, Adv. Math. 206 (2006), no. 1, 1–56. MR 2261750, DOI https://doi.org/10.1016/j.aim.2005.08.004
- Gennadi Kasparov and Guoliang Yu, The Novikov conjecture and geometry of Banach spaces, Geom. Topol. 16 (2012), no. 3, 1859–1880. MR 2980001, DOI https://doi.org/10.2140/gt.2012.16.1859
- Vincent Lafforgue, Un renforcement de la propriété (T), Duke Math. J. 143 (2008), no. 3, 559–602 (French, with English and French summaries). MR 2423763, DOI https://doi.org/10.1215/00127094-2008-029
- J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964), 263–287.
- J. Lindenstrauss, D. Preiss, and J. Tišer, Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces, Annals of Mathematics Studies, vol. 179, Princeton University Press, Princeton, NJ, 2012.
- Nathan Linial, Finite metric-spaces—combinatorics, geometry and algorithms, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 573–586. MR 1957562
- Nathan Linial, Eran London, and Yuri Rabinovich, The geometry of graphs and some of its algorithmic applications, Combinatorica 15 (1995), no. 2, 215–245. MR 1337355, DOI https://doi.org/10.1007/BF01200757
- J. Matoušek, On embedding expanders into $l_p$ spaces, Israel J. Math. 102 (1997), 189–197.
- Jiří Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR 1899299
- Bernard Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces $L^{p}$, Société Mathématique de France, Paris, 1974 (French). With an English summary; Astérisque, No. 11. MR 0344931
- Bernard Maurey, Type, cotype and $K$-convexity, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1299–1332. MR 1999197, DOI https://doi.org/10.1016/S1874-5849%2803%2980037-2
- S. Mazur and S. Ulam, Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946–948.
- Manor Mendel and Assaf Naor, Scaled Enflo type is equivalent to Rademacher type, Bull. Lond. Math. Soc. 39 (2007), no. 3, 493–498. MR 2331580, DOI https://doi.org/10.1112/blms/bdm016
- Manor Mendel and Assaf Naor, Metric cotype, Ann. of Math. (2) 168 (2008), no. 1, 247–298. MR 2415403, DOI https://doi.org/10.4007/annals.2008.168.247
- Manor Mendel and Assaf Naor, Nonlinear spectral calculus and super-expanders, Publ. Math. Inst. Hautes Études Sci. 119 (2014), 1–95. MR 3210176, DOI https://doi.org/10.1007/s10240-013-0053-2
- Assaf Naor, An introduction to the Ribe program, Jpn. J. Math. 7 (2012), no. 2, 167–233. MR 2995229, DOI https://doi.org/10.1007/s11537-012-1222-7
- Assaf Naor, Quantitative geometry, Proc. Natl. Acad. Sci. USA 110 (2013), no. 48, 19202–19205. MR 3153946, DOI https://doi.org/10.1073/pnas.1320388110
- A. Naor and G. Schechtman, Metric ${X}_p$ inequalities (2014), available at arXiv:1408.5819.
- A. Naor and G. Schechtman, Pythagorean powers of hypercubes (2015), available at arXiv:1501.05213.
- Piotr W. Nowak and Guoliang Yu, Large scale geometry, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2012. MR 2986138
- M. I. Ostrovskii, Embeddability of locally finite metric spaces into Banach spaces is finitely determined, Proc. Amer. Math. Soc. 140 (2012), no. 8, 2721–2730. MR 2910760, DOI https://doi.org/10.1090/S0002-9939-2011-11272-3
- Narutaka Ozawa, A note on non-amenability of ${\scr B}(l_p)$ for $p=1,2$, Internat. J. Math. 15 (2004), no. 6, 557–565. MR 2078880, DOI https://doi.org/10.1142/S0129167X04002430
- J. Pelant, Embeddings into $c_ 0$, Topology Appl. 57 (1994), 259–269.
- M. Ribe, On uniformly homeomorphic normed spaces, Ark. Mat. 14 (1976), no. 2, 237–244. MR 440340, DOI https://doi.org/10.1007/BF02385837
- M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Israel J. Math. 48 (1984), no. 2-3, 139–147. MR 770696, DOI https://doi.org/10.1007/BF02761159
- H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247–262. MR 611763, DOI https://doi.org/10.4064/fm-111-3-247-262
- Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR 1728880, DOI https://doi.org/10.1007/s002229900032
Review Information:
Reviewer: Florent P. Baudier
Affiliation: Department of Mathematics, Texas A&M University
Email: florent@math.tamu.edu
Reviewer: William B. Johnson
Affiliation: Department of Mathematics, Texas A&M University
Email: johnson@math.tamu.edu
Journal: Bull. Amer. Math. Soc. 53 (2016), 495-506
DOI: https://doi.org/10.1090/bull/1523
Published electronically: February 2, 2016
Review copyright: © Copyright 2016 American Mathematical Society