Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access July 1, 2015

Tight Embeddability of Proper and Stable Metric Spaces

  • F. Baudier and G. Lancien

Abstract

We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p ∈ [1,∞], every proper subset of Lp is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the ℓpn’s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.

MSC: 46B85; 46B20

References

[1] Y. Bartal and L. Gottlieb. Dimension reduction techniques for ℓp, 1 ℓ p < ∞, with applications. arXiv:1408.1789, 2014. Search in Google Scholar

[2] F. Baudier. Quantitative nonlinear embeddings into Lebesgue sequence spaces. J. Topol. Anal., to appear, arXiv: 1210.0588 (2013), 32 pages. Search in Google Scholar

[3] F. Baudier. Metrical characterization of super-reflexivity and linear type of Banach spaces. Arch. Math., 89:419–429, 2007. 10.1007/s00013-007-2108-4Search in Google Scholar

[4] F. Baudier. Plongements des espaces métriques dans les espaces de Banach. Ph.D. Thesis, 100 pages (French), 2009. Search in Google Scholar

[5] F. Baudier. Embeddings of proper metric spaces into Banach spaces. Hous. J. Math., 38:209–223, 2012. Search in Google Scholar

[6] F. Baudier and G. Lancien. Embeddings of locally finite metric spaces into Banach spaces. Proc. Amer. Math. Soc., 136: 1029–1033, 2008. 10.1090/S0002-9939-07-09109-5Search in Google Scholar

[7] Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications. American Mathematical Society, 2000. 10.1090/coll/048Search in Google Scholar

[8] W. J. Davis, T. Figiel,W. B. Johnson, and A. Pełczynski. Factoring weakly compact operators. J. Funct. Anal., 17:311–327, 1974. 10.1016/0022-1236(74)90044-5Search in Google Scholar

[9] P. Embrechts and M. Hofert. A note on generalized inverses. Math. Methods Oper. Res., 77(3):423–432, 2013. 10.1007/s00186-013-0436-7Search in Google Scholar

[10] D. J. H. Garling. Stable Banach spaces, random measures and Orlicz function spaces. In Probability measures on groups (Oberwolfach, 1981), volume 928 of Lecture Notes in Math., pages 121–175. Springer, Berlin-New York, 1982. 10.1007/BFb0093223Search in Google Scholar

[11] G. Godefroy and N. J. Kalton. Lipschitz-free Banach spaces. Studia Math., 159:121–141, 2003. 10.4064/sm159-1-6Search in Google Scholar

[12] 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. Search in Google Scholar

[13] E. Guentner and J. Kaminker. Exactness and uniform embeddability of discrete groups. J. London Math. Soc. (2), 70(3): 703–718, 2004. 10.1112/S0024610704005897Search in Google Scholar

[14] S. Heinrich and P. Mankiewicz. Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Studia Math., 73:225–251, 1982. 10.4064/sm-73-3-225-251Search in Google Scholar

[15] R. C. James. Uniformly non-square Banach spaces. Ann. of Math. (2), 80:542–550, 1964. 10.2307/1970663Search in Google Scholar

[16] W. B. Johnson and J. Lindenstrauss, editors. Handbook of the geometry of Banach spaces. Vol. I. North-Holland Publishing Co., Amsterdam, 2001. 10.1016/S1874-5849(01)80003-6Search in Google Scholar

[17] N. J. Kalton. Coarse and uniform embeddings into reflexive spaces. Quart. J. Math. (Oxford), 58:393–414, 2007. 10.1093/qmath/ham018Search in Google Scholar

[18] N. J. Kalton and N. L. Randrianarivony. The coarse Lipschitz structure of `p `q. Math. Ann., 341:223–237, 2008. 10.1007/s00208-007-0190-3Search in Google Scholar

[19] J. L. Krivine. Sous-espaces de dimension finie des espaces de Banach réticulés. Ann. of Math. (2), 104:1–29, 1976. 10.2307/1971054Search in Google Scholar

[20] J. L. Krivine and B. Maurey. Espaces de Banach stables. Israel J. Math., 39:273–295, 1981. 10.1007/BF02761674Search in Google Scholar

[21] Å. Lima, O. Nygaard, and E. Oja. Isometric factorization of weakly compact operators and the approximation property. Israel J. Math., 119:325–348, 2000. 10.1007/BF02810673Search in Google Scholar

[22] Y. Lindenstrauss and L. Tzafriri. Classical Banach Spaces, volume 338 of Lecture Notes Math. Springer-Verlag, 1979. 10.1007/978-3-662-35347-9Search in Google Scholar

[23] A. I. Markushevich. On a basis in a wide sense for linear spaces. Dokl. Akad. Nauk., 41:241–244, 1943. Search in Google Scholar

[24] B.Maurey and G. Pisier. Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Studia Math., 58:45–90, 1976. 10.4064/sm-58-1-45-90Search in Google Scholar

[25] P. Nowak and G. Yu. Large scale geometry. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2012. Search in Google Scholar

[26] M. I. Ostrovskii. Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Proc. Amer. Math. Soc., 140(8):2721–2730, 2012. 10.1090/S0002-9939-2011-11272-3Search in Google Scholar

[27] R. Ostrovsky and Y. Rabani. Polynomial-time approximation schemes for geometric min-sum median clustering. J. ACM, 49 (2):139–156 (electronic), 2002. 10.1145/506147.506149Search in Google Scholar

[28] M. Ribe. On uniformly homeomorphic normed spaces. Ark. Mat., 14:237–244, 1976. 10.1007/BF02385837Search in Google Scholar

[29] J. H. Wells and L. R. Williams. Embeddings and extensions in analysis. Springer-Verlag, New York-Heidelberg, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Search in Google Scholar

Received: 2015-3-20
Accepted: 2015-5-27
Published Online: 2015-7-1

© 2015 F. Baudier, G. Lancien

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 6.6.2023 from https://www.degruyter.com/document/doi/10.1515/agms-2015-0010/html
Scroll to top button