Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T05:51:05.765Z Has data issue: false hasContentIssue false
  • English
  • Français

DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  06 February 2018

S. OSTROVSKA  and
M. I. OSTROVSKII
S. OSTROVSKA
Affiliation:
Department of Mathematics, Atilim University, 06830 Incek, Ankara, Turkey e-mail: [email protected]
M. I. OSTROVSKII
Affiliation:
Department of Mathematics and Computer Science, St. John's University, 8000 Utopia Parkway, Queens, NY 11439, USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given a Banach space X and a real number α ≥ 1, we write: (1) D(X) ≤ α if, for any locally finite metric space A, all finite subsets of which admit bilipschitz embeddings into X with distortions ≤ C, the space A itself admits a bilipschitz embedding into X with distortion ≤ α ⋅ C; (2) D(X) = α+ if, for every ϵ > 0, the condition D(X) ≤ α + ϵ holds, while D(X) ≤ α does not; (3) D(X) ≤ α+ if D(X) = α+ or D(X) ≤ α. It is known that D(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D((⊕n=1∞Xn)p) ≤ 1+ for every nested family of finite-dimensional Banach spaces {Xn}n=1∞ and every 1 ≤ p ≤ ∞. (2) D((⊕n=1∞ ℓ∞n)p) = 1+ for 1 < p < ∞. (3) D(X) ≤ 4+ for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).

MSC classification

Primary: 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 1 , January 2019 , pp. 33 - 47
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Baudier, F., Embeddings of proper metric spaces into Banach spaces, Houston J. Math. 38 (1) (2012), 209–223.Google Scholar
2. Baudier, F. and Lancien, G., Embeddings of locally finite metric spaces into Banach spaces, Proc. Amer. Math. Soc. 136 (2008), 1029–1033.Google Scholar
3. Buyalo, S. and Schroeder, V., Elements of asymptotic geometry, EMS monographs in mathematics (European Mathematical Society, Zürich, 2007).Google Scholar
4. Fréchet, M., Les dimensions d'un ensemble abstrait, Math. Ann. 68 (3) (1910), 145–168.Google Scholar
5. Kalton, N. J. and Lancien, G., Best constants for Lipschitz embeddings of metric spaces into c 0, Fund. Math. 199 (2008), 249–272.Google Scholar
6. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces. I. Sequence spaces, Ergebnisse der mathematik und ihrer grenzgebiete, vol. 92 (Springer-Verlag, Berlin, 1977).Google Scholar
7. Linial, N., Finite metric spaces–combinatorics, geometry and algorithms, in Proceedings of the International Congress of Mathematicians, vol. III (Higher Education Press, Beijing, 2002), 573–586.Google Scholar
8. Matoušek, J., Lectures on Discrete Geometry, Graduate texts in mathematics, vol. 212. (Springer-Verlag, New York, 2002).Google Scholar
9. Maurey, B. and Pisier, G., Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Stud. Math. 58 (1) (1976), 45–90.Google Scholar
10. Naor, A., L 1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry, in Proceedings of the International Congress of Mathematicians, 2010, vol III (Hyderabad, India, 2011), 1549–1575.Google Scholar
11. Naor, A. and Peres, Y., Lp compression, traveling salesmen, and stable walks. Duke Math. J. 157 (1) (2011), 53–108.Google Scholar
12. Ostrovskii, M. I., Coarse embeddability into Banach spaces, Topol. Proc. 33 (2009), 163–183.Google Scholar
13. Ostrovskii, M. I., Embeddability of locally finite metric spaces into Banach spaces is finitely determined, Proc. Amer. Math. Soc. 140 (2012), 2721–2730.Google Scholar
14. Ostrovskii, M. I., Metric embeddings: Bilipschitz and coarse embeddings into Banach spaces, de Gruyter studies in mathematics, vol. 49 (Walter de Gruyter & Co., Berlin, 2013).Google Scholar
15. Williamson, D. P. and Shmoys, D. B., The design of approximation algorithms (Cambridge University Press, New York, NY, 2011).Google Scholar