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


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 


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  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

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