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

OSTROVSKA, S. ; OSTROVSKII, M. I.

Glasgow mathematical journal, Tome 61 (2019) no. 1, pp. 33-47

Résumé

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).

Détail

DOI : 10.1017/S0017089518000022

Bibliographie

[1] 1. Baudier, F., Embeddings of proper metric spaces into Banach spaces, Houston J. Math. 38 (1) (2012), 209–223. Google Scholar

[2] 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] 3. Buyalo, S. and Schroeder, V., Elements of asymptotic geometry, EMS monographs in mathematics (European Mathematical Society, Zürich, 2007). Google Scholar

[4] 4. Fréchet, M., Les dimensions d'un ensemble abstrait, Math. Ann. 68 (3) (1910), 145–168. Google Scholar

[5] 5. Kalton, N. J. and Lancien, G., Best constants for Lipschitz embeddings of metric spaces into c , Fund. Math. 199 (2008), 249–272. Google Scholar

[6] 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] 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] 8. Matoušek, J., Lectures on Discrete Geometry, Graduate texts in mathematics, vol. 212. (Springer-Verlag, New York, 2002). Google Scholar

[9] 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] 10. Naor, A., L 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] 11. Naor, A. and Peres, Y., L compression, traveling salesmen, and stable walks. Duke Math. J. 157 (1) (2011), 53–108. Google Scholar

[12] 12. Ostrovskii, M. I., Coarse embeddability into Banach spaces, Topol. Proc. 33 (2009), 163–183. Google Scholar

[13] 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] 14. Ostrovskii, M. I., Metric embeddings: Bilipschitz and coarse embeddings into Banach spaces, de Gruyter studies in mathematics, vol. 49 (Walter de Gruyter &amp; Co., Berlin, 2013). Google Scholar

[15] 15. Williamson, D. P. and Shmoys, D. B., The design of approximation algorithms (Cambridge University Press, New York, NY, 2011). Google Scholar