Israel Journal of Mathematics

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

On dominatedl 1 metrics



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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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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