American Mathematical Society

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.

MathSciNet review: 3497795
Full text of review: PDF This review is available free of charge.
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 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 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 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 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 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 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 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, DOI 10.1090/coll/048

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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, DOI 10.1007/978-1-4613-0039-7

Bernard Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces $L^{p}$, Astérisque, No. 11, Société Mathématique de France, Paris, 1974 (French). With an English summary. 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 10.1016/S1874-5849(03)80037-2

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 10.1112/blms/bdm016

Manor Mendel and Assaf Naor, Metric cotype, Ann. of Math. (2) 168 (2008), no. 1, 247–298. MR 2415403, DOI 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 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 10.1007/s11537-012-1222-7

Assaf Naor, Quantitative geometry, Proc. Natl. Acad. Sci. USA 110 (2013), no. 48, 19202–19205. MR 3153946, DOI 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, DOI 10.4171/112

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 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 10.1142/S0129167X04002430

J. Pelant, Embeddings into $c\sb 0$, Topology Appl. 57 (1994), 259–269.

M. Ribe, On uniformly homeomorphic normed spaces, Ark. Mat. 14 (1976), no. 2, 237–244. MR 440340, DOI 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 10.1007/BF02761159

H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247–262. MR 611763, DOI 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 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