Advertisement

Mathematische Annalen

, Volume 350, Issue 4, pp 953–972 | Cite as

Random embedding of \({\ell_p^n}\) into \({\ell_r^N}\)

  • Omer Friedland
  • Olivier GuédonEmail author
Article

Abstract

For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 < r < p < 2 with r ≤ 1, the operator \({S_r = S : \ell_p^n \to \ell_r^N}\) satisfies with overwhelming probability that \({\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}\), where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.

Mathematics Subject Classification (2000)

Primary 60E07 46B20 46B09 Secondary 52A21 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bernués J., López-Valdes M.: Tail estimates and a random embedding of \({l_p^n}\) into \({l_r^{(1+\epsilon)n}, 0 < r < p < 2}\) . Publ. Math. Debrecen 70(1–2), 9–18 (2007)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bourgain, J., Kalton, N.J., Tzafriri, L.: Geometry of finite-dimensional subspaces and quotients of L p. Geometric aspects of functional analysis (1987–88), pp. 138–175. Lecture Notes in Mathematics, 1376, Springer, Berlin (1989)Google Scholar
  3. 3.
    Bourgain J., Lindenstrauss J., Milman V.: Approximation of zonoids by zonotopes. Acta. Math. 162(1–2), 73–141 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bretagnolle J., Dacunha-Castelle D.: Application de l’étude de certaines formes linéaires aléatoires au plongement d’espaces de Banach dans des espaces L p. Ann. Sci. Ecole Norm. Sup. 4(2), 437–480 (1969)MathSciNetGoogle Scholar
  5. 5.
    Bretagnolle J., Dacunha-Castelle D., Krivine J.L.: Lois stables et espaces L p. Ann. Inst. H. Poincaré 2, 231–259 (1966)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Dvoretzky, A.: Some results on convex bodies and Banach spaces. In: 1961 Proceedings International Symposium of Linear Spaces (Jerusalem, 1960), pp. 123–160. Jerusalem Academic Press, Jerusalem; Pergamon, Oxford (1960)Google Scholar
  7. 7.
    Elton J.: Sign-embeddings of \({l^{n}_{1}}\) . Trans. Am. Math. Soc. 279(1), 113–124 (1983)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Esseen C.G.: On the concentration function of a sum of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9, 290–308 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Friedland, O., Guédon, O.: A multidimensional Esseen type inequality (work in progress)Google Scholar
  10. 10.
    Friedland O., Sodin S.: Bounds on the concentration function in terms of the Diophantine approximation. C. R. Math. Acad. Sci. Paris 345(9), 513–518 (2007)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Halász G.: Estimates for the concentration function of combinatorial number theory and probability. Period. Math. Hungar. 8(3–4), 197–211 (1977)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Johnson W.B., Schechtman G.: Embedding \({l^{m}_{p}}\) into \({l^{n}_{1}}\) . Acta. Math. 149(1–2), 71–85 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Johnson W.B., Schechtman G.: Very tight embeddings of subspaces of L p, 1 ≤ p < 2, into \({l^n_p}\) . Geom. Funct. Anal. 13(4), 845–851 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Johnson, W.B., Schechtman, G.: Finite dimensional subspaces of L p. Handbook of the Geometry of Banach Spaces, vol. I, pp. 837–870. Amsterdam, North-Holland (2001)Google Scholar
  15. 15.
    Kashin, B.S.: The widths of certain finite-dimensional sets and classes of smooth functions (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 41(2):334–351, 478 (1977)Google Scholar
  16. 16.
    Ledoux, M., Talagrand, M.: Probability in Banach spaces, pp. xii+480. Springer, Berlin (1991)Google Scholar
  17. 17.
    LePage R., Woodroofe M., Zinn J.: Convergence to a stable distribution via order statistics. Ann. Probab. 9(4), 624–632 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Litvak A.E., Pajor A., Rudelson M., Tomczak-Jaegermann N.: Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195, 491–523 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Marcus M.B., Pisier G.: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta. Math. 152(3–4), 245–301 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Maurey, B.: Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces L p (in French with an English summary), pp. ii+163. Astérisque No. 11. Société Mathématique de France, Paris (1974)Google Scholar
  21. 21.
    Maurey, B.: Type, cotype and K-convexity. Handbook of the geometry of Banach spaces, vol. 2, pp. 1299–1332. North-Holland, Amsterdam (2003)Google Scholar
  22. 22.
    Maurey B., Pisier G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach (French). Stud. Math. 58(1), 45–90 (1976)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Milman V.D.: A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies (Russian). Funkcional. Anal. i Prilozen. 5(4), 28–37 (1971)MathSciNetGoogle Scholar
  24. 24.
    Naor A., Zvavitch A.: Isomorphic embedding of \({l^n_p}\) , 1 < p < 2, into \({l^{(1+\epsilon)n}_1}\) . Isr. J. Math. 122, 371–380 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Pisier G.: On the dimension of the \({l^{n}_{p}}\) -subspaces of Banach spaces, for 1 ≤ p < 2. Trans. Am. Math. Soc. 276(1), 201–211 (1983)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Rosenthal H.P.: On subspaces of L p. Ann. Math. 97, 344–373 (1973)CrossRefGoogle Scholar
  27. 27.
    Rudelson M.: Invertibility of random matrices: norm of the inverse. Ann. Math. 168, 575–600 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Rudelson, M., Vershynin, R.: The smallest singular value of a random rectangular matrix. Commun. Pure Appl. Math. (2010) (to appear)Google Scholar
  29. 29.
    Schechtman G.: Special orthogonal splittings of \({L_1^{2k}}\) . Isr. J. Math. 189, 337–347 (2004)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Szarek S.: On Kashin’s almost Euclidean orthogonal decomposition of \({l^{1}_{n}}\) . Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26(8), 691–694 (1978)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Szarek S., Tomczak-Jaegermann N.: On nearly Euclidean decomposition for some classes of Banach spaces. Compos. Math. 40(3), 367–385 (1980)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Talagrand M.: Embedding subspaces of L 1 into \({\ell_1^N}\) . Proc. Am. Math. Soc. 108(2), 363–369 (1990)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Talagrand, M.: Embedding subspaces of L p in \({l^n_p}\) . Geometric aspects of functional analysis (Israel, 1992–1994). Oper. Theory Adv. Appl., vol. 77, pp. 311–325. Birkhäuser, Basel (1995)Google Scholar
  34. 34.
    Tao, T., Vu, V.: Additive combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105, pp. xviii+512. Cambridge University Press, Cambridge (2006)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité Pierre et Marie Curie, Paris 6ParisFrance
  2. 2.Université Paris-Est, Équipe d’Analyse et Mathématiques AppliquéesMarne-la-Vallée Cedex 2France

Personalised recommendations