Mathematics – Functional Analysis
Scientific paper
2011-02-24
Comptes rendus de l'Academie bulgare des Sciences, 64 (2011), no. 6, 775-784
Mathematics
Functional Analysis
Scientific paper
The paper contains the following results and observations: (1) There exists a sequence of unweighted graphs $\{G_n\}_n$ with maximum degree 3 such that a Banach space $X$ has no nontrivial cotype iff $\{G_n\}_n$ admit uniformly bilipschitz embeddings into $X$; (2) The same for Banach spaces with no nontrivial type; (3) A sequence $\{G_n\}$ characterizing Banach spaces with no nontrivial cotype in the sense described above can be chosen to be a sequence of bounded degree expanders; (4) The infinite diamond does not admit a bilipschitz embedding into Banach spaces with the Radon-Nikod\'{y}m property; (5) A new proof of the Cheeger-Kleiner result: The Laakso space does not admit a bilipschitz embedding into Banach spaces with the Radon-Nikod\'{y}m property; (6) A new proof of the Johnson-Schechtman result: uniform bilipschitz embeddability of finite diamonds into a Banach space implies its nonsuperreflexivity.
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