Advertisement

Geometriae Dedicata

, Volume 172, Issue 1, pp 387–398 | Cite as

A doubling subset of \(L_p\) for \(p>2\) that is inherently infinite dimensional

  • Vincent Lafforgue
  • Assaf NaorEmail author
Original Paper

Abstract

It is shown that for every \(p\in (2,\infty )\) there exists a doubling subset of \(L_p\) that does not admit a bi-Lipschitz embedding into \(\mathbb R^k\) for any \(k\in \mathbb N\).

Keywords

Metric embeddings Heisenberg group Doubling metric spaces 

Mathematics Subject Classification

30L05 20F65 46E30 

Notes

Acknowledgments

V. L. was supported by ANR Grant KInd. A. N. was supported by NSF Grant CCF-0832795, BSF Grant 2010021, the Packard Foundation and the Simons Foundation. Part of this work was completed while A. N. was a Visiting Fellow at Princeton University.

References

  1. 1.
    Lang, U., Plaut, C.: Bilipschitz embeddings of metric spaces into space forms. Geom. Dedicata 87, 285–307 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Naor, A., Neiman, O.: Assouad’s theorem with dimension independent of the snowflaking. Rev. Mat. Iberoam. 28, 1123–1142 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bartal, Y., Gottlieb, A., Neiman, O.: On the impossibility of dimension reduction for doubling subsets of \(\ell _p,\, p>2\). Preprint available at http://arxiv.org/abs/1308.4996, (2013)
  4. 4.
    Lee, J.R., Mendel, M., Naor, A.: Metric structures in L\(_1\): dimension, snowflakes and average distortion. Eur. J. Combin. 26, 1180–1190 (2005)Google Scholar
  5. 5.
    Laakso, T.J.: Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality. Geom. Funct. Anal. 10, 111–123 (2000)Google Scholar
  6. 6.
    Gupta, A., Newman, I., Rabinovich, Y., Sinclair, A.: Cuts, trees and L\(_1\): embeddings of graphs. Combinatorica 24, 233–269 (2004)Google Scholar
  7. 7.
    Mendel, M., Naor, A.: Markov convexity and local rigidity of distorted metrics [extended abstract]. In: Teillaud, M. (ed.) Computational Geometry (SCG’08), pp. 49–58. ACM, New York (2008)Google Scholar
  8. 8.
    Ostrovskii, M.I.: On metric characterizations of some classes of Banach spaces. C. R. Acad. Bulgare Sci. 64, 775–784 (2011)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Blachère, S.: Word distance on the discrete Heisenberg group. Colloq. Math. 95, 21–36 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cheeger, J., Kleiner, B., Naor, A.: Compression bounds for Lipschitz maps from the Heisenberg group to L\(_1\). Acta Math. 207, 291–373 (2011)Google Scholar
  11. 11.
    Austin, T., Naor, A., Tessera, R.:Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces. Groups Geom. Dyn. 7(3) 497–522 (2013)Google Scholar
  12. 12.
    Lafforgue, V., Naor, A.: Vertical versus horizontal Poincaré inequalities on the Heisenberg group. To appear in Israel J. Math., preprint available at http://arxiv.org/abs/1212.2107, (2012)
  13. 13.
    Lee, J.R., Naor, A.: \({L}_p\) metrics on the Heisenberg group and the Goemans-Linial conjecture. In: Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 99–108, (2006) Available at http://www.cims.nyu.edu/~naor/homepage%20files/L_pHGL.pdf
  14. 14.
    Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44, 522–536 (1938)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Semmes, S.: An introduction to Heisenberg groups in analysis and geometry. Notices Am. Math. Soc. 50, 640–646 (2003)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Cygan, J.: Subadditivity of homogeneous norms on certain nilpotent Lie groups. Proc. Am. Math. Soc. 83, 69–70 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Coifman, R.R., and Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Mathematics, Vol. 242. Springer, Berlin (1971) Étude de certaines intégrales singulièresGoogle Scholar
  18. 18.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)Google Scholar
  19. 19.
    Wojtaszczyk, P.: Banach Spaces for Analysts, Volume 25 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques—Analyse, Probabilités, Modélisation—Orléans (MAPMO) UMR CNRS 6628Université d’Orléans Rue de ChartresOrléans cedex 2France
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations