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

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  06 February 2018

S. OSTROVSKA  and
M. I. OSTROVSKII
S. OSTROVSKA
Affiliation:
Department of Mathematics, Atilim University, 06830 Incek, Ankara, Turkey e-mail: sofia.ostrovska@atilim.edu.tr
M. I. OSTROVSKII
Affiliation:
Department of Mathematics and Computer Science, St. John's University, 8000 Utopia Parkway, Queens, NY 11439, USA e-mail: ostrovsm@stjohns.edu
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Abstract

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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∞ ℓ∞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).

MSC classification

Primary: 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 1 , January 2019 , pp. 33 - 47
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

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