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2016 | OriginalPaper | Buchkapitel

Metric Characterizations of Some Classes of Banach Spaces

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Abstract

The main purpose of the paper is to present some recent results on metric characterizations of superreflexivity and the Radon–Nikodým property.

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Literatur
1.
Zurück zum Zitat J.M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, H. Short, Notes on word hyperbolic groups. Edited by Short, in Group Theory from a Geometrical Viewpoint (Trieste, 1990) (World Scientific, River Edge, NJ, 1991), pp. 3–63 J.M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, H. Short, Notes on word hyperbolic groups. Edited by Short, in Group Theory from a Geometrical Viewpoint (Trieste, 1990) (World Scientific, River Edge, NJ, 1991), pp. 3–63
2.
Zurück zum Zitat N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces. Stud. Math. 57 (2), 147–190 (1976)MathSciNetMATH N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces. Stud. Math. 57 (2), 147–190 (1976)MathSciNetMATH
3.
Zurück zum Zitat K. Ball, The Ribe programme, in Séminaire Bourbaki, vol. 2011/2012. Exposés 1043–1058. Astérisque, vol. 352 (2013). Exp. No. 1047, viii, 147–159 K. Ball, The Ribe programme, in Séminaire Bourbaki, vol. 2011/2012. Exposés 1043–1058. Astérisque, vol. 352 (2013). Exp. No. 1047, viii, 147–159
4.
Zurück zum Zitat F. Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces. Archiv. Math. 89 (5), 419–429 (2007)MathSciNetCrossRefMATH F. Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces. Archiv. Math. 89 (5), 419–429 (2007)MathSciNetCrossRefMATH
5.
Zurück zum Zitat F. Baudier, Embeddings of proper metric spaces into Banach spaces. Houst. J. Math. 38 (1), 209–223 (2012)MathSciNetMATH F. Baudier, Embeddings of proper metric spaces into Banach spaces. Houst. J. Math. 38 (1), 209–223 (2012)MathSciNetMATH
6.
Zurück zum Zitat F. Baudier, G. Lancien, Embeddings of locally finite metric spaces into Banach spaces. Proc. Am. Math. Soc. 136, 1029–1033 (2008)MathSciNetCrossRefMATH F. Baudier, G. Lancien, Embeddings of locally finite metric spaces into Banach spaces. Proc. Am. Math. Soc. 136, 1029–1033 (2008)MathSciNetCrossRefMATH
7.
9.
10.
Zurück zum Zitat I. Benjamini, O. Schramm, Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal. 7 (3), 403–419 (1997)MathSciNetCrossRefMATH I. Benjamini, O. Schramm, Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal. 7 (3), 403–419 (1997)MathSciNetCrossRefMATH
11.
Zurück zum Zitat Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1. American Mathematical Society Colloquium Publications, vol. 48 (American Mathematical Society, Providence, RI, 2000) Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1. American Mathematical Society Colloquium Publications, vol. 48 (American Mathematical Society, Providence, RI, 2000)
12.
Zurück zum Zitat R.D. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Lecture Notes in Mathematics, vol. 993 (Springer, Berlin, 1983) R.D. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Lecture Notes in Mathematics, vol. 993 (Springer, Berlin, 1983)
13.
14.
Zurück zum Zitat J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, in Geometrical Aspects of Functional Analysis (1985/86). Lecture Notes in Mathematics, vol. 1267 (Springer, Berlin, 1987), pp. 157–167 J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, in Geometrical Aspects of Functional Analysis (1985/86). Lecture Notes in Mathematics, vol. 1267 (Springer, Berlin, 1987), pp. 157–167
15.
16.
17.
Zurück zum Zitat J. Bourgain, La propriété de Radon-Nikodym Publications mathématiques de l’Université Pierre et Marie Curie, Paris 36, 104 (1979) J. Bourgain, La propriété de Radon-Nikodym Publications mathématiques de l’Université Pierre et Marie Curie, Paris 36, 104 (1979)
18.
Zurück zum Zitat M.R. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319 (Springer, Berlin, 1999) M.R. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319 (Springer, Berlin, 1999)
19.
Zurück zum Zitat N. Brown, E. Guentner, Uniform embeddings of bounded geometry spaces into reflexive Banach space. Proc. Am. Math. Soc. 133 (7), 2045–2050 (2005)MathSciNetCrossRefMATH N. Brown, E. Guentner, Uniform embeddings of bounded geometry spaces into reflexive Banach space. Proc. Am. Math. Soc. 133 (7), 2045–2050 (2005)MathSciNetCrossRefMATH
20.
Zurück zum Zitat A. Brunel, L. Sucheston, On J-convexity and some ergodic super-properties of Banach spaces. Trans. Am. Math. Soc. 204, 79–90 (1975)MathSciNetMATH A. Brunel, L. Sucheston, On J-convexity and some ergodic super-properties of Banach spaces. Trans. Am. Math. Soc. 204, 79–90 (1975)MathSciNetMATH
21.
22.
Zurück zum Zitat S. Buyalo, A. Dranishnikov, V. Schroeder, Embedding of hyperbolic groups into products of binary trees. Invent. Math. 169 (1), 153–192 (2007)MathSciNetCrossRefMATH S. Buyalo, A. Dranishnikov, V. Schroeder, Embedding of hyperbolic groups into products of binary trees. Invent. Math. 169 (1), 153–192 (2007)MathSciNetCrossRefMATH
23.
Zurück zum Zitat L. Capogna, D. Danielli, S.D. Pauls, J.T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Progress in Mathematics, vol. 259 (Birkhäuser, Basel, 2007) L. Capogna, D. Danielli, S.D. Pauls, J.T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Progress in Mathematics, vol. 259 (Birkhäuser, Basel, 2007)
24.
25.
26.
Zurück zum Zitat J. Cheeger, B. Kleiner, On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces, in Inspired by S.S. Chern. Nankai Tracts in Mathematics, vol. 11 (World Scientific, Hackensack, NJ, 2006), 129–152 J. Cheeger, B. Kleiner, On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces, in Inspired by S.S. Chern. Nankai Tracts in Mathematics, vol. 11 (World Scientific, Hackensack, NJ, 2006), 129–152
27.
Zurück zum Zitat J. Cheeger, B. Kleiner, Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property. Geom. Funct. Anal. 19 (4), 1017–1028 (2009)MathSciNetCrossRefMATH J. Cheeger, B. Kleiner, Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property. Geom. Funct. Anal. 19 (4), 1017–1028 (2009)MathSciNetCrossRefMATH
28.
Zurück zum Zitat J. Cheeger, B. Kleiner, Differentiating maps into L 1, and the geometry of BV functions. Ann. Math. (2) 171 (2), 1347–1385 (2010). arXiv:math.MG/0611954 J. Cheeger, B. Kleiner, Differentiating maps into L 1, and the geometry of BV functions. Ann. Math. (2) 171 (2), 1347–1385 (2010). arXiv:math.MG/0611954
29.
Zurück zum Zitat J. Cheeger, B. Kleiner, A. Naor, Compression bounds for Lipschitz maps from the Heisenberg group to L 1. Acta Math. 207, 291–373 (2011)MathSciNetCrossRefMATH J. Cheeger, B. Kleiner, A. Naor, Compression bounds for Lipschitz maps from the Heisenberg group to L 1. Acta Math. 207, 291–373 (2011)MathSciNetCrossRefMATH
30.
Zurück zum Zitat J.P.R. Christensen, Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings. Publ. Dép. Math. (Lyon) 10 (2), 29–39 (1973). Actes du Deuxième Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1973), I, pp. 29–39. J.P.R. Christensen, Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings. Publ. Dép. Math. (Lyon) 10 (2), 29–39 (1973). Actes du Deuxième Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1973), I, pp. 29–39.
32.
Zurück zum Zitat H. Corson, V. Klee, Topological classification of convex sets, in Proceedings of Symposia in Pure Mathematics, vol. VII, pp. 37–51 (American Mathematical Society, Providence, RI, 1963) H. Corson, V. Klee, Topological classification of convex sets, in Proceedings of Symposia in Pure Mathematics, vol. VII, pp. 37–51 (American Mathematical Society, Providence, RI, 1963)
33.
Zurück zum Zitat W.J. Davis, J. Lindenstrauss, The ℓ 1 n problem and degrees of non-reflexivity. II. Stud. Math. 58 (2), 179–196 (1976)MathSciNetMATH W.J. Davis, J. Lindenstrauss, The 1 n problem and degrees of non-reflexivity. II. Stud. Math. 58 (2), 179–196 (1976)MathSciNetMATH
34.
Zurück zum Zitat W.J. Davis, W.B. Johnson, J. Lindenstrauss, The ℓ 1 n problem and degrees of non-reflexivity. Stud. Math. 55 (2), 123–139 (1976)MathSciNetMATH W.J. Davis, W.B. Johnson, J. Lindenstrauss, The 1 n problem and degrees of non-reflexivity. Stud. Math. 55 (2), 123–139 (1976)MathSciNetMATH
35.
Zurück zum Zitat Y. de Cornulier, R. Tessera, A. Valette, Isometric group actions on Hilbert spaces: growth of cocycles. Geom. Funct. Anal. 17 (3), 770–792 (2007)MathSciNetCrossRefMATH Y. de Cornulier, R. Tessera, A. Valette, Isometric group actions on Hilbert spaces: growth of cocycles. Geom. Funct. Anal. 17 (3), 770–792 (2007)MathSciNetCrossRefMATH
36.
Zurück zum Zitat J. Diestel, J.J. Uhl, Jr. Vector measures, in With a Foreword by B.J. Pettis. Mathematical Surveys, vol. 15 (American Mathematical Society, Providence, RI, 1977) J. Diestel, J.J. Uhl, Jr. Vector measures, in With a Foreword by B.J. Pettis. Mathematical Surveys, vol. 15 (American Mathematical Society, Providence, RI, 1977)
37.
Zurück zum Zitat S.J. Dilworth, D. Kutzarova, G. Lancien, N.L. Randrianarivony, Asymptotic geometry of Banach spaces and uniform quotient maps. Proc. Am. Math. Soc. 142 (8), 2747–2762 (2014)MathSciNetCrossRefMATH S.J. Dilworth, D. Kutzarova, G. Lancien, N.L. Randrianarivony, Asymptotic geometry of Banach spaces and uniform quotient maps. Proc. Am. Math. Soc. 142 (8), 2747–2762 (2014)MathSciNetCrossRefMATH
38.
Zurück zum Zitat P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm. Isr. J. Math. 13, 281–288 (1972)MathSciNetCrossRef P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm. Isr. J. Math. 13, 281–288 (1972)MathSciNetCrossRef
39.
Zurück zum Zitat P. Enflo, Uniform homeomorphisms between Banach spaces. Séminaire d’Analyse Fonctionnelle (1975–1976), Exp. No. 18, 6 pp., École Polytechnique, Palaiseau, 1976 P. Enflo, Uniform homeomorphisms between Banach spaces. Séminaire d’Analyse Fonctionnelle (1975–1976), Exp. No. 18, 6 pp., École Polytechnique, Palaiseau, 1976
40.
Zurück zum Zitat P. Enflo, On infinite-dimensional topological groups. Séminaire sur la Géométrie des Espaces de Banach (1977–1978), Exp. No. 10–11, 11 pp., École Polytechnique, Palaiseau, 1978 P. Enflo, On infinite-dimensional topological groups. Séminaire sur la Géométrie des Espaces de Banach (1977–1978), Exp. No. 10–11, 11 pp., École Polytechnique, Palaiseau, 1978
42.
Zurück zum Zitat O. Giladi, A. Naor, Improved bounds in the scaled Enflo type inequality for Banach spaces. Extracta Math. 25 (2), 151–164 (2010)MathSciNetMATH O. Giladi, A. Naor, Improved bounds in the scaled Enflo type inequality for Banach spaces. Extracta Math. 25 (2), 151–164 (2010)MathSciNetMATH
43.
Zurück zum Zitat O. Giladi, M. Mendel, A. Naor, Improved bounds in the metric cotype inequality for Banach spaces. J. Funct. Anal. 260, 164–194 (2011)MathSciNetCrossRefMATH O. Giladi, M. Mendel, A. Naor, Improved bounds in the metric cotype inequality for Banach spaces. J. Funct. Anal. 260, 164–194 (2011)MathSciNetCrossRefMATH
44.
Zurück zum Zitat O. Giladi, A. Naor, G. Schechtman, Bourgain’s discretization theorem. Annales Mathematiques de la faculte des sciences de Toulouse, XXI (4), 817–837 (2012) O. Giladi, A. Naor, G. Schechtman, Bourgain’s discretization theorem. Annales Mathematiques de la faculte des sciences de Toulouse, XXI (4), 817–837 (2012)
45.
46.
Zurück zum Zitat M. Gromov, Hyperbolic groups, in Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8 (Springer, New York, 1987), pp. 75–263 M. Gromov, Hyperbolic groups, in Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8 (Springer, New York, 1987), pp. 75–263
47.
Zurück zum Zitat M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian Geometry, ed. by A. Bellaïche, J.-J. Risler. Progress in Mathematics, vol. 144 (Birkhäuser, Basel, 1996), pp. 79–323 M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian Geometry, ed. by A. Bellaïche, J.-J. Risler. Progress in Mathematics, vol. 144 (Birkhäuser, Basel, 1996), pp. 79–323
49.
Zurück zum Zitat A. Gupta, Steiner points in tree metrics don’t (really) help, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Washington, DC, 2001 (SIAM, Philadelphia, PA, 2001), pp. 220–227 A. Gupta, Steiner points in tree metrics don’t (really) help, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Washington, DC, 2001 (SIAM, Philadelphia, PA, 2001), pp. 220–227
50.
Zurück zum Zitat A. Gupta, I. Newman, Y. Rabinovich, A. Sinclair, Cuts, trees and ℓ 1-embeddings of graphs. Combinatorica 24, 233–269 (2004). Conference version in 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp. 399–408 A. Gupta, I. Newman, Y. Rabinovich, A. Sinclair, Cuts, trees and 1-embeddings of graphs. Combinatorica 24, 233–269 (2004). Conference version in 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp. 399–408
51.
Zurück zum Zitat J.M. Heinonen, Lectures on Analysis on Metric Spaces. Universitext (Springer, New York, 2001)CrossRefMATH J.M. Heinonen, Lectures on Analysis on Metric Spaces. Universitext (Springer, New York, 2001)CrossRefMATH
52.
Zurück zum Zitat S. Heinrich, P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Stud. Math. 73 (3), 225–251 (1982)MathSciNetMATH S. Heinrich, P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Stud. Math. 73 (3), 225–251 (1982)MathSciNetMATH
55.
Zurück zum Zitat R.C. James, Some self-dual properties of normed linear spaces, in Symposium on Infinite-Dimensional Topology, Louisiana State University, Baton Rouge, LA, 1967). Annals of Mathematics Studies, vol. 69 (Princeton University Press, Princeton, NJ, 1972), pp. 159–175 R.C. James, Some self-dual properties of normed linear spaces, in Symposium on Infinite-Dimensional Topology, Louisiana State University, Baton Rouge, LA, 1967). Annals of Mathematics Studies, vol. 69 (Princeton University Press, Princeton, NJ, 1972), pp. 159–175
57.
Zurück zum Zitat R.C. James, Structure of Banach spaces: Radon-Nikodým and other properties, in General Topology and Modern Analysis. Proceedings of Conference, University of California, Riverside, CA, 1980 (Academic Press, New York, London, 1981), pp. 347–363 R.C. James, Structure of Banach spaces: Radon-Nikodým and other properties, in General Topology and Modern Analysis. Proceedings of Conference, University of California, Riverside, CA, 1980 (Academic Press, New York, London, 1981), pp. 347–363
60.
Zurück zum Zitat B. Kleiner, J. Mackay, Differentiable structures on metric measure spaces: a primer. arXiv:1108.1324 B. Kleiner, J. Mackay, Differentiable structures on metric measure spaces: a primer. arXiv:1108.1324
62.
Zurück zum Zitat S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Stud. Math. 44, 583–595 (1972)MathSciNetMATH S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Stud. Math. 44, 583–595 (1972)MathSciNetMATH
63.
Zurück zum Zitat T.J. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincare inequality. Geom. Funct. Anal. 10 (1), 111–123 (2000)MathSciNetCrossRefMATH T.J. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincare inequality. Geom. Funct. Anal. 10 (1), 111–123 (2000)MathSciNetCrossRefMATH
64.
Zurück zum Zitat G. Lancien, A short course on nonlinear geometry of Banach spaces, in Topics in Functional and Harmonic Analysis. Theta Series in Advanced Mathematics, vol. 14 (Theta, Bucharest, 2013), pp. 77–101 G. Lancien, A short course on nonlinear geometry of Banach spaces, in Topics in Functional and Harmonic Analysis. Theta Series in Advanced Mathematics, vol. 14 (Theta, Bucharest, 2013), pp. 77–101
65.
66.
Zurück zum Zitat J.R. Lee, A. Naor, L p metrics on the Heisenberg group and the Goemans-Linial conjecture, in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (IEEE, Piscataway, NJ, 2006), pp. 99–108 J.R. Lee, A. Naor, L p metrics on the Heisenberg group and the Goemans-Linial conjecture, in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (IEEE, Piscataway, NJ, 2006), pp. 99–108
67.
Zurück zum Zitat J.R. Lee, A. Naor, Y. Peres, Trees and Markov convexity. Geom. Funct. Anal. 18, 1609–1659 (2009). Conference version: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1028–1037 (ACM, New York, 2006), arXiv:0706.0545 J.R. Lee, A. Naor, Y. Peres, Trees and Markov convexity. Geom. Funct. Anal. 18, 1609–1659 (2009). Conference version: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1028–1037 (ACM, New York, 2006), arXiv:0706.0545
68.
Zurück zum Zitat S. Li, Coarse differentiation and quantitative nonembeddability for Carnot groups. J. Funct. Anal. 266, 4616–4704 (2014)MathSciNetCrossRefMATH S. Li, Coarse differentiation and quantitative nonembeddability for Carnot groups. J. Funct. Anal. 266, 4616–4704 (2014)MathSciNetCrossRefMATH
69.
Zurück zum Zitat S. Li, Markov convexity and nonembeddability of the Heisenberg group. Ann. Inst. Fourier, to appear, arXiv: 1404.6751 S. Li, Markov convexity and nonembeddability of the Heisenberg group. Ann. Inst. Fourier, to appear, arXiv: 1404.6751
70.
Zurück zum Zitat J. Lindenstrauss, A. Pełczyński, Absolutely summing operators in \(\mathcal{L}_{p}\)-spaces and their applications. Studia Math. 29, 275–326 (1968)MathSciNetMATH J. Lindenstrauss, A. Pełczyński, Absolutely summing operators in \(\mathcal{L}_{p}\)-spaces and their applications. Studia Math. 29, 275–326 (1968)MathSciNetMATH
71.
Zurück zum Zitat N. Linial, E. London, Y. Rabinovich, The geometry of graphs and some of its algorithmic applications. Combinatorica 15 (2), 215–245 (1995)MathSciNetCrossRefMATH N. Linial, E. London, Y. Rabinovich, The geometry of graphs and some of its algorithmic applications. Combinatorica 15 (2), 215–245 (1995)MathSciNetCrossRefMATH
72.
Zurück zum Zitat P. Mankiewicz, On the differentiability of Lipschitz mappings in Fréchet spaces. Stud. Math. 45, 15–29 (1973)MATH P. Mankiewicz, On the differentiability of Lipschitz mappings in Fréchet spaces. Stud. Math. 45, 15–29 (1973)MATH
74.
Zurück zum Zitat J. Matoušek, Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212 (Springer, New York, 2002) J. Matoušek, Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212 (Springer, New York, 2002)
75.
Zurück zum Zitat S. Mazur, S. Ulam, Sur les transformationes isométriques d’espaces vectoriels normés. C. R. Acad. Sci. Paris 194, 946–948 (1932)MATH S. Mazur, S. Ulam, Sur les transformationes isométriques d’espaces vectoriels normés. C. R. Acad. Sci. Paris 194, 946–948 (1932)MATH
76.
78.
Zurück zum Zitat M. Mendel, A. Naor, Markov convexity and local rigidity of distorted metrics. J. Eur. Math. Soc. 15 (1), 287–337 (2013). Conference version: Computational geometry (SCG’08) (ACM, New York, 2008), pp. 49–58 M. Mendel, A. Naor, Markov convexity and local rigidity of distorted metrics. J. Eur. Math. Soc. 15 (1), 287–337 (2013). Conference version: Computational geometry (SCG’08) (ACM, New York, 2008), pp. 49–58
79.
Zurück zum Zitat D.P. Milman, V.D. Milman, The geometry of nested families with empty intersection. Structure of the unit sphere of a nonreflexive space (Russian). Matem. Sbornik 66 (1), 109–118 (1965). English transl.: Am. Math. Soc. Transl. (2) 85, 233–243 (1969) D.P. Milman, V.D. Milman, The geometry of nested families with empty intersection. Structure of the unit sphere of a nonreflexive space (Russian). Matem. Sbornik 66 (1), 109–118 (1965). English transl.: Am. Math. Soc. Transl. (2) 85, 233–243 (1969)
80.
Zurück zum Zitat R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications. Mathematical Surveys and Monographs, vol. 91 (American Mathematical Society, Providence, RI, 2002) R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications. Mathematical Surveys and Monographs, vol. 91 (American Mathematical Society, Providence, RI, 2002)
82.
Zurück zum Zitat P.W. Nowak, G. Yu, Large Scale Geometry. EMS Textbooks in Mathematics (European Mathematical Society, Zürich, 2012) P.W. Nowak, G. Yu, Large Scale Geometry. EMS Textbooks in Mathematics (European Mathematical Society, Zürich, 2012)
83.
Zurück zum Zitat D. Osajda, Small cancellation labellings of some infinite graphs and applications. arXiv:1406.5015 D. Osajda, Small cancellation labellings of some infinite graphs and applications. arXiv:1406.5015
84.
Zurück zum Zitat M.I. Ostrovskii, Finite dimensional characteristics related to superreflexivity of Banach spaces. Bull. Aust. Math. Soc. 69, 289–295 (2004)MathSciNetCrossRefMATH M.I. Ostrovskii, Finite dimensional characteristics related to superreflexivity of Banach spaces. Bull. Aust. Math. Soc. 69, 289–295 (2004)MathSciNetCrossRefMATH
85.
Zurück zum Zitat M.I. Ostrovskii, Coarse embeddings of locally finite metric spaces into Banach spaces without cotype. C. R. Acad. Bulg. Sci. 59 (11), 1113–1116 (2006)MathSciNetMATH M.I. Ostrovskii, Coarse embeddings of locally finite metric spaces into Banach spaces without cotype. C. R. Acad. Bulg. Sci. 59 (11), 1113–1116 (2006)MathSciNetMATH
86.
Zurück zum Zitat M.I. Ostrovskii, Coarse embeddability into Banach spaces, Topol. Proc. 33, 163–183 (2009)MathSciNetMATH M.I. Ostrovskii, Coarse embeddability into Banach spaces, Topol. Proc. 33, 163–183 (2009)MathSciNetMATH
87.
Zurück zum Zitat M.I. Ostrovskii, On metric characterizations of some classes of Banach spaces. C. R. Acad. Bulg. Sci. 64 (6), 775–784 (2011)MathSciNetMATH M.I. Ostrovskii, On metric characterizations of some classes of Banach spaces. C. R. Acad. Bulg. Sci. 64 (6), 775–784 (2011)MathSciNetMATH
88.
Zurück zum Zitat M.I. Ostrovskii, Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Proc. Am. Math. Soc. 140, 2721–2730 (2012)MathSciNetCrossRefMATH M.I. Ostrovskii, Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Proc. Am. Math. Soc. 140, 2721–2730 (2012)MathSciNetCrossRefMATH
89.
Zurück zum Zitat M.I. Ostrovskii, Different forms of metric characterizations of classes of Banach spaces. Houst. J. Math. 39 (3), 889–906 (2013)MathSciNetMATH M.I. Ostrovskii, Different forms of metric characterizations of classes of Banach spaces. Houst. J. Math. 39 (3), 889–906 (2013)MathSciNetMATH
90.
Zurück zum Zitat M.I. Ostrovskii, Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces. de Gruyter Studies in Mathematics, vol. 49 (Walter de Gruyter, Berlin, 2013) M.I. Ostrovskii, Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces. de Gruyter Studies in Mathematics, vol. 49 (Walter de Gruyter, Berlin, 2013)
91.
Zurück zum Zitat M.I. Ostrovskii, On metric characterizations of the Radon-Nikodým and related properties of Banach spaces. J. Topol. Anal. 6 (3), 441–464 (2014)MathSciNetCrossRefMATH M.I. Ostrovskii, On metric characterizations of the Radon-Nikodým and related properties of Banach spaces. J. Topol. Anal. 6 (3), 441–464 (2014)MathSciNetCrossRefMATH
92.
Zurück zum Zitat M.I. Ostrovskii, Metric characterizations of superreflexivity in terms of word hyperbolic groups and finite graphs. Anal. Geom. Metr. Spaces 2, 154–168 (2014)MathSciNetMATH M.I. Ostrovskii, Metric characterizations of superreflexivity in terms of word hyperbolic groups and finite graphs. Anal. Geom. Metr. Spaces 2, 154–168 (2014)MathSciNetMATH
93.
Zurück zum Zitat M.I. Ostrovskii, Metric spaces nonembeddable into Banach spaces with the Radon-Nikodým property and thick families of geodesics. Fundam. Math. 227, 85–95 (2014)CrossRefMATH M.I. Ostrovskii, Metric spaces nonembeddable into Banach spaces with the Radon-Nikodým property and thick families of geodesics. Fundam. Math. 227, 85–95 (2014)CrossRefMATH
94.
95.
Zurück zum Zitat M.I. Ostrovskii, Test-space characterizations of some classes of Banach spaces, in Algebraic Methods in Functional Analysis ed. by I.G. Todorov, L. Turowska. The Victor Shulman Anniversary Volume. Operator Theory: Advances and Applications, vol. 233 (Birkhäuser, Basel, 2014), pp. 103–126 M.I. Ostrovskii, Test-space characterizations of some classes of Banach spaces, in Algebraic Methods in Functional Analysis ed. by I.G. Todorov, L. Turowska. The Victor Shulman Anniversary Volume. Operator Theory: Advances and Applications, vol. 233 (Birkhäuser, Basel, 2014), pp. 103–126
98.
Zurück zum Zitat P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. (2) 129 (1), 1–60 (1989) P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. (2) 129 (1), 1–60 (1989)
99.
Zurück zum Zitat A. Pełczyński, A note on the paper of I. Singer “Basic sequences and reflexivity of Banach spaces”. Stud. Math. 21, 371–374 (1961/1962) A. Pełczyński, A note on the paper of I. Singer “Basic sequences and reflexivity of Banach spaces”. Stud. Math. 21, 371–374 (1961/1962)
100.
Zurück zum Zitat J.C.B. Perrott, Transfinite duals of Banach spaces and ergodic super-properties equivalent to super-reflexivity. Q. J. Math. Oxford Ser. (2) 30 (117), 99–111 (1979) J.C.B. Perrott, Transfinite duals of Banach spaces and ergodic super-properties equivalent to super-reflexivity. Q. J. Math. Oxford Ser. (2) 30 (117), 99–111 (1979)
102.
Zurück zum Zitat G. Pisier, Probabilistic methods in the geometry of Banach spaces, in Probability and Analysis (Varenna, 1985). Lecture Notes in Mathematics, vol. 1206 (Springer, Berlin, 1986), pp. 167–241 G. Pisier, Probabilistic methods in the geometry of Banach spaces, in Probability and Analysis (Varenna, 1985). Lecture Notes in Mathematics, vol. 1206 (Springer, Berlin, 1986), pp. 167–241
103.
Zurück zum Zitat G. Pisier, Martingales in Banach spaces. Cambridge Studies in Advanced Mathematics, vol. 155 (Cambridge University Press, Cambridge, 2016) G. Pisier, Martingales in Banach spaces. Cambridge Studies in Advanced Mathematics, vol. 155 (Cambridge University Press, Cambridge, 2016)
104.
Zurück zum Zitat V. Pták, Biorthogonal systems and reflexivity of Banach spaces. Czechoslov. Math. J. 9, 319–326 (1959)MathSciNetMATH V. Pták, Biorthogonal systems and reflexivity of Banach spaces. Czechoslov. Math. J. 9, 319–326 (1959)MathSciNetMATH
105.
Zurück zum Zitat Y. Rabinovich, R. Raz, Lower bounds on the distortion of embedding finite metric spaces in graphs. Discret. Comput. Geom. 19 (1), 79–94 (1998)MathSciNetCrossRefMATH Y. Rabinovich, R. Raz, Lower bounds on the distortion of embedding finite metric spaces in graphs. Discret. Comput. Geom. 19 (1), 79–94 (1998)MathSciNetCrossRefMATH
107.
Zurück zum Zitat M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces. Isr. J. Math. 48 (2–3), 139–147 (1984)MathSciNetCrossRefMATH M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces. Isr. J. Math. 48 (2–3), 139–147 (1984)MathSciNetCrossRefMATH
108.
Zurück zum Zitat J.J. Schäffer, K. Sundaresan, Reflexivity and the girth of spheres. Math. Ann. 184, 163–168 (1969/1970) J.J. Schäffer, K. Sundaresan, Reflexivity and the girth of spheres. Math. Ann. 184, 163–168 (1969/1970)
109.
Zurück zum Zitat S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A ∞ -weights. Rev. Mat. Iberoamericana 12 (2), 337–410 (1996)MathSciNetCrossRefMATH S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A -weights. Rev. Mat. Iberoamericana 12 (2), 337–410 (1996)MathSciNetCrossRefMATH
110.
Zurück zum Zitat I. Singer, Basic sequences and reflexivity of Banach spaces. Stud. Math. 21, 351–369 (1961/1962) I. Singer, Basic sequences and reflexivity of Banach spaces. Stud. Math. 21, 351–369 (1961/1962)
111.
Zurück zum Zitat C. Stegall, The Radon-Nikodým property in conjugate Banach spaces. Trans. Am. Math. Soc. 206, 213–223 (1975)CrossRefMATH C. Stegall, The Radon-Nikodým property in conjugate Banach spaces. Trans. Am. Math. Soc. 206, 213–223 (1975)CrossRefMATH
113.
Zurück zum Zitat D. van Dulst, The geometry of Banach spaces with the Radon-Nikodým property. Rend. Circ. Mat. Palermo (2) Suppl. 7, ix+81 pp (1985). D. van Dulst, The geometry of Banach spaces with the Radon-Nikodým property. Rend. Circ. Mat. Palermo (2) Suppl. 7, ix+81 pp (1985).
114.
Zurück zum Zitat J. Wenzel, Superreflexivity and J-convexity of Banach spaces. Acta Math. Univ. Comenian. (N.S.) 66, 135–147 (1997) J. Wenzel, Superreflexivity and J-convexity of Banach spaces. Acta Math. Univ. Comenian. (N.S.) 66, 135–147 (1997)
Metadaten
Titel
Metric Characterizations of Some Classes of Banach Spaces
verfasst von
Mikhail Ostrovskii
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-30961-3_15