Israel Journal of Mathematics

, Volume 123, Issue 1, pp 285–301 | Cite as

On dominatedl 1 metrics

  • Jiří Matoušek
  • Yuri Rabinovich


We introduce and study a classl 1 dom (ρ) ofl 1-embeddable metrics corresponding to a given metric ρ. This class is defined as the set of all convex combinations of ρ-dominated line metrics. Such metrics were implicitly used before in several constuctions of low-distortion embeddings intol p -spaces, such as Bourgain’s embedding of an arbitrary metric ρ onn points withO(logh) distortion. Our main result is that the gap between the distortions of embedding of a finite metric ρ of sizen intol 2 versus intol 1 dom (ρ) is at most\(O\left( {\sqrt {\log n} } \right)\), and that this bound is essentially tight. A significant part of the paper is devoted to proving lower bounds on distortion of such embeddings. We also discuss some general properties and concrete examples.


Convex Combination Isoperimetric Inequality Line Metrics Complete Binary Tree Finite Subspace 
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  1. [1]
    N. Alon and J. Spencer,The Probabilistic Method, Wiley, New York, 1992.MATHGoogle Scholar
  2. [2]
    Y. Bartal,On approximating arbitrary metrics by tree metrics, inProceeding of the 30th Annual ACM Symposium on Theory of Computing, ACM Press New York, 1998, pp. 161–168.Google Scholar
  3. [3]
    [3] B. Bollobás,Martingales, isoperimetric inequalities and random graphs, in52. Combinatorics, Eger (Hungary), Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam, 1987, pp. 113–139.Google Scholar
  4. [4]
    J. Bourgain,On Lipschitz embedding of finite metric spaces in Hilbert space, Israel Journal of Mathematics52 (1985), 46–52.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    J. Bourgain,The metrical interpretation of superreflexivity in Banach spaces, Israel Journal of Mathematics56 (1986), 222–230.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    P. Enflo,On a problem of Smirnov, Arkiv för Matematik8 (1969), 107–109.CrossRefMathSciNetGoogle Scholar
  7. [7]
    P. Enflo,On the nonexistence of uniform homeomorphisms between L p-spaces, Arkiv för Matematik8 (1969), 103–105.CrossRefMathSciNetGoogle Scholar
  8. [8]
    P. Erdős and C. A. Rogers,Covering space with convex bodies, Acta Arithmetica7 (1962), 281–285.MathSciNetGoogle Scholar
  9. [9]
    U. Feige,Approximating the bandwidth via volume respecting embeddings, inProceedings of the 30th Annual ACM Symposium on Theory of Computing, ACM Press New York, 1998, pp 90–99.Google Scholar
  10. [10]
    D. Gillman, R. Permantel and Y. Rabinovich,An inverse Chernoff bound, manuscript, Haifa University, 1999; submitted to Information Processing Letters.Google Scholar
  11. [11]
    M. Gromov,Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, Boston, 1999.MATHGoogle Scholar
  12. [12]
    A. Gupta, I. Newman, A. Sinclair and Y. Rabinovich,Cuts, trees and l 1 embeddings of graphs, inProceedings of the 40th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1999, pp. 399–408.Google Scholar
  13. [13]
    L. H. Harper,Optimal numberings and isoperimetric problems on graphs, Journal of Combinatorial Theory1 (1966), 385–393.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    W. B. Johnson and J. Lindenstrauss,Extentions of Lipschitz mappings into a Hilbert space, Contemporary Mathematics26 (1984), 189–206.MATHMathSciNetGoogle Scholar
  15. [15]
    N. Linial, E. London and Y. Rabinovich,The geometry of graphs and some of its algorithmic applications, Combinatorica15 (1995), 215–245.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    J. Matoušek,On embedding trees into uniformly convex Banach spaces, Israel Journal of Mathematics114 (1999), 221–237.MathSciNetMATHGoogle Scholar
  17. [17]
    V. Milman and G. Schechtman,Asymptomic Theory of Finite Dimensional Spaces, Lecture Notes in Mathematics1200, Springer-Verlag, Berlin, 1986.Google Scholar
  18. [18]
    S. B. Rao,Small distortion and volume preserving embeddings for planar and Euclidean metrics, inProceedings of the 15th Annual ACM Symposium on Computational Geometry, ACM Press, New York, NY, 1999, pp. 300–306.Google Scholar
  19. [19]
    C. A. Rogers,Covering a sphere with spheres, Mathematika10 (1963), 157–164.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    J. H. Wells and L. R. Williams,Embeddings and Extensions in Analysis, Springer, Berlin, 1975.MATHGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2001

Authors and Affiliations

  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic
  2. 2.CS DepartmentHaifa UniversityHaifaIsrael

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