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DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES

  • S. OSTROVSKA (a1) and M. I. OSTROVSKII (a2)
Abstract

Given a Banach space X and a real number α ≥ 1, we write: (1) D(X) ≤ α if, for any locally finite metric space A, all finite subsets of which admit bilipschitz embeddings into X with distortions ≤ C, the space A itself admits a bilipschitz embedding into X with distortion ≤ α ⋅ C; (2) D(X) = α+ if, for every ϵ > 0, the condition D(X) ≤ α + ϵ holds, while D(X) ≤ α does not; (3) D(X) ≤ α+ if D(X) = α+ or D(X) ≤ α. It is known that D(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D((⊕ n=1 Xn ) p ) ≤ 1+ for every nested family of finite-dimensional Banach spaces {Xn } n=1 and every 1 ≤ p ≤ ∞. (2) D((⊕ n=1 n ) p ) = 1+ for 1 < p < ∞. (3) D(X) ≤ 4+ for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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