DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES
Glasgow mathematical journal, Tome 61 (2019) no. 1, pp. 33-47
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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).
OSTROVSKA, S.; OSTROVSKII, M. I. DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES. Glasgow mathematical journal, Tome 61 (2019) no. 1, pp. 33-47. doi: 10.1017/S0017089518000022
@article{10_1017_S0017089518000022,
author = {OSTROVSKA, S. and OSTROVSKII, M. I.},
title = {DISTORTION {IN} {THE} {FINITE} {DETERMINATION} {RESULT} {FOR} {EMBEDDINGS} {OF} {LOCALLY} {FINITE} {METRIC} {SPACES} {INTO} {BANACH} {SPACES}},
journal = {Glasgow mathematical journal},
pages = {33--47},
year = {2019},
volume = {61},
number = {1},
doi = {10.1017/S0017089518000022},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089518000022/}
}
TY - JOUR AU - OSTROVSKA, S. AU - OSTROVSKII, M. I. TI - DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES JO - Glasgow mathematical journal PY - 2019 SP - 33 EP - 47 VL - 61 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.1017/S0017089518000022/ DO - 10.1017/S0017089518000022 ID - 10_1017_S0017089518000022 ER -
%0 Journal Article %A OSTROVSKA, S. %A OSTROVSKII, M. I. %T DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES %J Glasgow mathematical journal %D 2019 %P 33-47 %V 61 %N 1 %U http://geodesic.mathdoc.fr/articles/10.1017/S0017089518000022/ %R 10.1017/S0017089518000022 %F 10_1017_S0017089518000022
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