This is an announcement for the paper "On metric characterizations of some classes of Banach spaces" by Mikhail Ostrovskii.
Abstract: 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.
Archive classification: math.FA math.MG
Mathematics Subject Classification: 46B85, Secondary: 05C12, 46B07, 46B22, 54E35
Submitted from: ostrovsm@stjohns.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1102.5082
or
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alspach＠math.okstate.edu